The use of general descriptive names, registered names, trademarks, service marks, etc. in this p... more The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use.
The development thus far in the previous chapters has not excluded the case p = 1, a situation wh... more The development thus far in the previous chapters has not excluded the case p = 1, a situation which almost always requires special treatment involving both analysis and geometry. Our objective in this chapter is to reformulate the Gaussian 1-capacity, characterize the Gaussian Poincare 1-inequality and Ehrhard’s inequality as well as the Gaussian isoperimetry, and handle the Gaussian ∞-capacity as the dual form of the Gaussian 1-capacity.
ANNALI SCUOLA NORMALE SUPERIORE - CLASSE DI SCIENZE, 2018
This paper shows that under 8 >>>>>>>< >>>>>>>: 0 &l... more This paper shows that under 8 >>>>>>>< >>>>>>>: 0 < , n; 1 < n; 1 p, q < 1; p 1(n ) < ↵ < min n, 1+ p 1 ; = p 1q( ↵p) + n < 8 >< >: + " 8 " > 0 as ↵ 1 p < 1 + " 8 " > 0 as 1 < p < ↵ 1 + (n )(n ↵ ) n ↵ as 1= p<↵ 1, if μ is a nonnegative Radon measure of finite -variation on Rn then the Morrey potential class I↵L p, embeds continuously into the Campanato class Lq, μ , and its converse also holds with μ being admissible. Mathematics Subject Classification (2010): 31C15 (primary); 42B35, 46E35 (secondary).
In this chapter we introduce the notion of the Gaussian p-capacity which is important at least in... more In this chapter we introduce the notion of the Gaussian p-capacity which is important at least in describing the appropriate tracing of the Gaussian Sobolev p-space.
This is a PDF file of an article that has undergone enhancements after acceptance, such as the ad... more This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Journal de Mathématiques Pures et Appliquées, 2020
The employment effects of environmental technologies are in the focus of politicians but there ar... more The employment effects of environmental technologies are in the focus of politicians but there are only few studies analyzing these effects for different environmental innovation fields. We use the 2009 wave of the German part of the Community Innovation Panel (CIS) allowing for such an analysis at the firm level. The main focus of the paper lies on the analysis of the adaptation behavior of firms with respect to the relationship of employment and (environmental) innovation. We use an endogenous switching regression approach to take the simultaneous character of innovation activities and employment demand into consideration. Our econometric analysis shows that innovative firms in general are characterized by a significantly more dynamic employment development. Especially the realization of environmental process innovations leads to a higher employment within the firm. The theoretical background of this finding is that process innovation induced cost savings improve the competitiveness of firms. This has a positive effect on demand and thus also increases employment. A more detailed analysis by different environmental innovation fields shows that material and energy savings are positively correlated to employment because they especially help to increase the profitability and competitiveness of the firm. On the other side, air and water process innovations that are still dominated by end-of-pipe technologies have a negative impact on the employment development.
It is well known that Lipschitz spaces on the torus are an algebra. It is no more the case in the... more It is well known that Lipschitz spaces on the torus are an algebra. It is no more the case in the non compact situation because of the behavior at infinity. This is a companion article to Bonami et al. (J Math Pures Appl (9) 131:130–170, 2019), where pointwise multipliers on Lipschitz spaces on $${\mathbb {R}}^n$$ are characterized for non-integer values of the parameter. In this article, the authors first establish two equivalent characterizations of a modified Zygmund space, and then characterize the pointwise multipliers on Lipschitz spaces on $${\mathbb {R}}^n$$ for the integer values of the parameter, in particular, for the Zygmund class, via the intersection of the Lebesgue space $$L^\infty ({\mathbb {R}}^n)$$ and the modified Zygmund space. This result can be used to show that the bilinear decomposition of the pointwise product of the Hardy space and its dual, in the integer values of the parameter, obtained in the aforementioned reference is sharp in the dual space sense.
This article shows that if HL p,κ and HL q,λ μ are (p, κ)-Hardy-Morrey space based on the standar... more This article shows that if HL p,κ and HL q,λ μ are (p, κ)-Hardy-Morrey space based on the standard Lebesgue measure and (q, λ)-Hardy-Morrey space induced by the non-negative Radon measure μ respectively then one has the following trace law for the Hardy-Morrey-Sobolev space I α (HL p,κ) of Riesz potentials of order α of HL p,κ-functions: sup
This paper shows that under ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 0 < s < 1; 0 < α < n; 0 < β ≤ n; 1 ≤ mi... more This paper shows that under ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 0 < s < 1; 0 < α < n; 0 < β ≤ n; 1 ≤ min{ p, q} ≤ max{ p, q} < β p(n − α p) −1 < ∞; λ = q(np −1 − s − α) + n − β, if μ is a nonnegative Radon measure on R n with the β-dimensional upper curvature |||μ||| β < ∞ then I αΛ p,∞ s (the mean Hölder-Lipschitz potential space on R n) embeds continuously into L q,λ μ (the curved Campanato-Radon space on R n); and yet its converse is still valid with μ being admissible/doubling (cf. Theorem 1.1), thereby discovering the γ-Hölder-Lipschitz continuity of any duality solution to the α-th Laplace equation (−Δ) α 2 u = μ or the [1, n 2) ∩ {1, 2,. .. , n} k-th Hessian equation F k [u] = μ under a suitable curvature condition |||μ||| β < ∞ (cf. Theorems 1.3 and 1.5). Communicated by Loukas Grafakos.
Calculus of Variations and Partial Differential Equations, 2019
Post-operative cognitive dysfunction (POCD) occurs after major surgery in elderly patients and af... more Post-operative cognitive dysfunction (POCD) occurs after major surgery in elderly patients and affects the quality of patients' lives. The present study aims to explore the protective effects and possible mechanisms of compound K in old mice with POCD caused by partial hepatectomy. Methods: Sixteen month-old mice were administered different doses of compound K from the 8th day to 14th day after partial hepatectomy. Cognitive function was subsequently measured with a Morris water-maze (MWM) test. Serum inflammatory cytokine levels were measured by magnetic bead panel; levels of cytokines in the hippocampus were analyzed using immunohistochemistry and immunoblotting. The mRNA levels of target genes were measured using real-time PCR. Results: Compared with the model group, MWM scores were significantly attenuated at days 10 and 14 postsurgery in mice receiving compound K (10, 30 mg/kg) in a dose-dependent manner. Both systemic and local cytokine levels were reduced after treatment of compound K. The alterations in serum lipids were independent of the attenuation of POCD syndrome. An inhibitor of liver X receptor-α (LXRα), GGPP, reversed the effects of compound K. Conclusions: The results provide evidence for an alleviation of POCD by compound K via local inflammation inhibition in hippocampus tissue; furthermore, the data suggests the mechanism involves the LXRα pathway. The present study supports further evaluation of compound K as a potential effective modulator for POCD.
In this article, the authors characterize the Morrey spaces as well as their preduals via quadrat... more In this article, the authors characterize the Morrey spaces as well as their preduals via quadratic functions related to the Taylor remainder of the kernel of the Riesz potential. As applications, the authors obtain some strong capacitary inequalities, which are then used to study the regularity of the duality/weak solution to the fractional Laplace equation with measure data.
Mathematical Methods in the Applied Sciences, 2018
The novelties appearing in these approaches exist in applications of the multiresolution analysis... more The novelties appearing in these approaches exist in applications of the multiresolution analysis of the wavelets on metric measure spaces of homogeneous type, the bilinear decomposition of the product space H 1 () × BMO(), the (sub)bilinear decomposition of commutators, the proof of off-diagonal estimates of the action of Calderón-Zygmund operators on the wavelet functions, and the boundedness of the almost diagonal matrix on the spaces H 1 () and BMO(). Notably, throughout this article, is not assumed to satisfy the reverse doubling condition.
Let (X, µ) be a measure space. In this paper, using some ideas from Grafakos and Kalton, the auth... more Let (X, µ) be a measure space. In this paper, using some ideas from Grafakos and Kalton, the authors establish an off-diagonal Marcinkiewicz interpolation theorem for a quasilinear operator T in Lorentz spaces L p,q (X) with p, q ∈ (0,∞], which is a corrected version of Theorem 1.4.19 in [Grafakos, L.: Classical Fourier Analysis, Second Edition, Graduate Texts in
through the tracing and duality laws based on Riesz's singular integral operator I s. We discover... more through the tracing and duality laws based on Riesz's singular integral operator I s. We discover that f ∈ I s [H s,1 − ] * if and only if ∃ g = (g 1 , ..., g n) ∈ L ∞ n such that f = R • g = n j=1 R j g j in BMO (the John-Nirenberg space introduced in their 1961 Comm. Pure Appl. Math. paper [14]) where R = (R 1 , ..., R n) is the vector-valued Riesz transform-this characterizes the Riesz transform part R • L ∞ n of Fefferman-Stein's decomposition (established in their 1972 Acta Math paper [9]) for BMO = L ∞ + R • L ∞ n and yet indicates that I s [H s,1 − ] * is indeed a solution to Bourgain-Brezis' problem under n ≥ 2: "What are the function spaces X, W 1,n ⊂ X ⊂ BMO, such that every F ∈ X has a decomposition F = n j=1 R j Y j where Y j ∈ L ∞ ?"
We prove an abstract form of Hardy's inequality for local and non-local regular Dirichlet forms o... more We prove an abstract form of Hardy's inequality for local and non-local regular Dirichlet forms on metric measure spaces, using the Green operator of the Dirichlet form in question. Under additional assumptions such as the volume doubling, the reverse volume doubling, and certain natural estimates of the Green function, we obtain the "classical" form of Hardy's inequality containing distance to a reference point or set.
A nonlinear superposition operator T g related to a Borel measurable function g : C → C is define... more A nonlinear superposition operator T g related to a Borel measurable function g : C → C is defined via T g (f) := g • f for any complex-valued function f on R n. This article is devoted to investigating the mapping properties of T g on a new BMO type space recently introduced by Bourgain, Brezis and Mironescu [J. Eur. Math. Soc. (JEMS) 17 (2015), 2083-2101], as well as its VMO and CMO type subspaces. Some sufficient and necessary conditions for the inclusion result and the continuity property of T g on these spaces are obtained.
Journal de Mathématiques Pures et Appliquées, 2019
For any p ∈ (0, 1) and α = 1/p − 1, let H p (R n) and C α (R n) be the Hardy and the Campanato sp... more For any p ∈ (0, 1) and α = 1/p − 1, let H p (R n) and C α (R n) be the Hardy and the Campanato spaces on the n-dimensional Euclidean space R n , respectively. In this article, the authors find suitable Musielak-Orlicz functions Φ p , defined by setting, for any x ∈ R n and t ∈ [0, ∞),
Assume that (X, d, µ) is a space of homogeneous type in the sense of Coifman and Weiss. In this a... more Assume that (X, d, µ) is a space of homogeneous type in the sense of Coifman and Weiss. In this article, motivated by the breakthrough work of P. Auscher and T. Hytönen on orthonormal bases of regular wavelets on spaces of homogeneous type, the authors introduce a new kind of approximations of the identity with exponential decay (for short, exp-ATI). Via such an exp-ATI, motivated by another creative idea of Y. Han et al. to merge the aforementioned orthonormal bases of regular wavelets into the frame of the existed distributional theory on spaces of homogeneous type, the authors establish the homogeneous continuous/discrete Calderón reproducing formulae on (X, d, µ), as well as their inhomogeneous counterparts. The novelty of this article exists in that d is only assumed to be a quasi-metric and the underlying measure µ a doubling measure, not necessary to satisfy the reverse doubling condition. It is well known that Calderón reproducing formulae are the cornerstone to develop analysis and, especially, harmonic analysis on spaces of homogeneous type.
The use of general descriptive names, registered names, trademarks, service marks, etc. in this p... more The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use.
The development thus far in the previous chapters has not excluded the case p = 1, a situation wh... more The development thus far in the previous chapters has not excluded the case p = 1, a situation which almost always requires special treatment involving both analysis and geometry. Our objective in this chapter is to reformulate the Gaussian 1-capacity, characterize the Gaussian Poincare 1-inequality and Ehrhard’s inequality as well as the Gaussian isoperimetry, and handle the Gaussian ∞-capacity as the dual form of the Gaussian 1-capacity.
ANNALI SCUOLA NORMALE SUPERIORE - CLASSE DI SCIENZE, 2018
This paper shows that under 8 >>>>>>>< >>>>>>>: 0 &l... more This paper shows that under 8 >>>>>>>< >>>>>>>: 0 < , n; 1 < n; 1 p, q < 1; p 1(n ) < ↵ < min n, 1+ p 1 ; = p 1q( ↵p) + n < 8 >< >: + " 8 " > 0 as ↵ 1 p < 1 + " 8 " > 0 as 1 < p < ↵ 1 + (n )(n ↵ ) n ↵ as 1= p<↵ 1, if μ is a nonnegative Radon measure of finite -variation on Rn then the Morrey potential class I↵L p, embeds continuously into the Campanato class Lq, μ , and its converse also holds with μ being admissible. Mathematics Subject Classification (2010): 31C15 (primary); 42B35, 46E35 (secondary).
In this chapter we introduce the notion of the Gaussian p-capacity which is important at least in... more In this chapter we introduce the notion of the Gaussian p-capacity which is important at least in describing the appropriate tracing of the Gaussian Sobolev p-space.
This is a PDF file of an article that has undergone enhancements after acceptance, such as the ad... more This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Journal de Mathématiques Pures et Appliquées, 2020
The employment effects of environmental technologies are in the focus of politicians but there ar... more The employment effects of environmental technologies are in the focus of politicians but there are only few studies analyzing these effects for different environmental innovation fields. We use the 2009 wave of the German part of the Community Innovation Panel (CIS) allowing for such an analysis at the firm level. The main focus of the paper lies on the analysis of the adaptation behavior of firms with respect to the relationship of employment and (environmental) innovation. We use an endogenous switching regression approach to take the simultaneous character of innovation activities and employment demand into consideration. Our econometric analysis shows that innovative firms in general are characterized by a significantly more dynamic employment development. Especially the realization of environmental process innovations leads to a higher employment within the firm. The theoretical background of this finding is that process innovation induced cost savings improve the competitiveness of firms. This has a positive effect on demand and thus also increases employment. A more detailed analysis by different environmental innovation fields shows that material and energy savings are positively correlated to employment because they especially help to increase the profitability and competitiveness of the firm. On the other side, air and water process innovations that are still dominated by end-of-pipe technologies have a negative impact on the employment development.
It is well known that Lipschitz spaces on the torus are an algebra. It is no more the case in the... more It is well known that Lipschitz spaces on the torus are an algebra. It is no more the case in the non compact situation because of the behavior at infinity. This is a companion article to Bonami et al. (J Math Pures Appl (9) 131:130–170, 2019), where pointwise multipliers on Lipschitz spaces on $${\mathbb {R}}^n$$ are characterized for non-integer values of the parameter. In this article, the authors first establish two equivalent characterizations of a modified Zygmund space, and then characterize the pointwise multipliers on Lipschitz spaces on $${\mathbb {R}}^n$$ for the integer values of the parameter, in particular, for the Zygmund class, via the intersection of the Lebesgue space $$L^\infty ({\mathbb {R}}^n)$$ and the modified Zygmund space. This result can be used to show that the bilinear decomposition of the pointwise product of the Hardy space and its dual, in the integer values of the parameter, obtained in the aforementioned reference is sharp in the dual space sense.
This article shows that if HL p,κ and HL q,λ μ are (p, κ)-Hardy-Morrey space based on the standar... more This article shows that if HL p,κ and HL q,λ μ are (p, κ)-Hardy-Morrey space based on the standard Lebesgue measure and (q, λ)-Hardy-Morrey space induced by the non-negative Radon measure μ respectively then one has the following trace law for the Hardy-Morrey-Sobolev space I α (HL p,κ) of Riesz potentials of order α of HL p,κ-functions: sup
This paper shows that under ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 0 < s < 1; 0 < α < n; 0 < β ≤ n; 1 ≤ mi... more This paper shows that under ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 0 < s < 1; 0 < α < n; 0 < β ≤ n; 1 ≤ min{ p, q} ≤ max{ p, q} < β p(n − α p) −1 < ∞; λ = q(np −1 − s − α) + n − β, if μ is a nonnegative Radon measure on R n with the β-dimensional upper curvature |||μ||| β < ∞ then I αΛ p,∞ s (the mean Hölder-Lipschitz potential space on R n) embeds continuously into L q,λ μ (the curved Campanato-Radon space on R n); and yet its converse is still valid with μ being admissible/doubling (cf. Theorem 1.1), thereby discovering the γ-Hölder-Lipschitz continuity of any duality solution to the α-th Laplace equation (−Δ) α 2 u = μ or the [1, n 2) ∩ {1, 2,. .. , n} k-th Hessian equation F k [u] = μ under a suitable curvature condition |||μ||| β < ∞ (cf. Theorems 1.3 and 1.5). Communicated by Loukas Grafakos.
Calculus of Variations and Partial Differential Equations, 2019
Post-operative cognitive dysfunction (POCD) occurs after major surgery in elderly patients and af... more Post-operative cognitive dysfunction (POCD) occurs after major surgery in elderly patients and affects the quality of patients' lives. The present study aims to explore the protective effects and possible mechanisms of compound K in old mice with POCD caused by partial hepatectomy. Methods: Sixteen month-old mice were administered different doses of compound K from the 8th day to 14th day after partial hepatectomy. Cognitive function was subsequently measured with a Morris water-maze (MWM) test. Serum inflammatory cytokine levels were measured by magnetic bead panel; levels of cytokines in the hippocampus were analyzed using immunohistochemistry and immunoblotting. The mRNA levels of target genes were measured using real-time PCR. Results: Compared with the model group, MWM scores were significantly attenuated at days 10 and 14 postsurgery in mice receiving compound K (10, 30 mg/kg) in a dose-dependent manner. Both systemic and local cytokine levels were reduced after treatment of compound K. The alterations in serum lipids were independent of the attenuation of POCD syndrome. An inhibitor of liver X receptor-α (LXRα), GGPP, reversed the effects of compound K. Conclusions: The results provide evidence for an alleviation of POCD by compound K via local inflammation inhibition in hippocampus tissue; furthermore, the data suggests the mechanism involves the LXRα pathway. The present study supports further evaluation of compound K as a potential effective modulator for POCD.
In this article, the authors characterize the Morrey spaces as well as their preduals via quadrat... more In this article, the authors characterize the Morrey spaces as well as their preduals via quadratic functions related to the Taylor remainder of the kernel of the Riesz potential. As applications, the authors obtain some strong capacitary inequalities, which are then used to study the regularity of the duality/weak solution to the fractional Laplace equation with measure data.
Mathematical Methods in the Applied Sciences, 2018
The novelties appearing in these approaches exist in applications of the multiresolution analysis... more The novelties appearing in these approaches exist in applications of the multiresolution analysis of the wavelets on metric measure spaces of homogeneous type, the bilinear decomposition of the product space H 1 () × BMO(), the (sub)bilinear decomposition of commutators, the proof of off-diagonal estimates of the action of Calderón-Zygmund operators on the wavelet functions, and the boundedness of the almost diagonal matrix on the spaces H 1 () and BMO(). Notably, throughout this article, is not assumed to satisfy the reverse doubling condition.
Let (X, µ) be a measure space. In this paper, using some ideas from Grafakos and Kalton, the auth... more Let (X, µ) be a measure space. In this paper, using some ideas from Grafakos and Kalton, the authors establish an off-diagonal Marcinkiewicz interpolation theorem for a quasilinear operator T in Lorentz spaces L p,q (X) with p, q ∈ (0,∞], which is a corrected version of Theorem 1.4.19 in [Grafakos, L.: Classical Fourier Analysis, Second Edition, Graduate Texts in
through the tracing and duality laws based on Riesz's singular integral operator I s. We discover... more through the tracing and duality laws based on Riesz's singular integral operator I s. We discover that f ∈ I s [H s,1 − ] * if and only if ∃ g = (g 1 , ..., g n) ∈ L ∞ n such that f = R • g = n j=1 R j g j in BMO (the John-Nirenberg space introduced in their 1961 Comm. Pure Appl. Math. paper [14]) where R = (R 1 , ..., R n) is the vector-valued Riesz transform-this characterizes the Riesz transform part R • L ∞ n of Fefferman-Stein's decomposition (established in their 1972 Acta Math paper [9]) for BMO = L ∞ + R • L ∞ n and yet indicates that I s [H s,1 − ] * is indeed a solution to Bourgain-Brezis' problem under n ≥ 2: "What are the function spaces X, W 1,n ⊂ X ⊂ BMO, such that every F ∈ X has a decomposition F = n j=1 R j Y j where Y j ∈ L ∞ ?"
We prove an abstract form of Hardy's inequality for local and non-local regular Dirichlet forms o... more We prove an abstract form of Hardy's inequality for local and non-local regular Dirichlet forms on metric measure spaces, using the Green operator of the Dirichlet form in question. Under additional assumptions such as the volume doubling, the reverse volume doubling, and certain natural estimates of the Green function, we obtain the "classical" form of Hardy's inequality containing distance to a reference point or set.
A nonlinear superposition operator T g related to a Borel measurable function g : C → C is define... more A nonlinear superposition operator T g related to a Borel measurable function g : C → C is defined via T g (f) := g • f for any complex-valued function f on R n. This article is devoted to investigating the mapping properties of T g on a new BMO type space recently introduced by Bourgain, Brezis and Mironescu [J. Eur. Math. Soc. (JEMS) 17 (2015), 2083-2101], as well as its VMO and CMO type subspaces. Some sufficient and necessary conditions for the inclusion result and the continuity property of T g on these spaces are obtained.
Journal de Mathématiques Pures et Appliquées, 2019
For any p ∈ (0, 1) and α = 1/p − 1, let H p (R n) and C α (R n) be the Hardy and the Campanato sp... more For any p ∈ (0, 1) and α = 1/p − 1, let H p (R n) and C α (R n) be the Hardy and the Campanato spaces on the n-dimensional Euclidean space R n , respectively. In this article, the authors find suitable Musielak-Orlicz functions Φ p , defined by setting, for any x ∈ R n and t ∈ [0, ∞),
Assume that (X, d, µ) is a space of homogeneous type in the sense of Coifman and Weiss. In this a... more Assume that (X, d, µ) is a space of homogeneous type in the sense of Coifman and Weiss. In this article, motivated by the breakthrough work of P. Auscher and T. Hytönen on orthonormal bases of regular wavelets on spaces of homogeneous type, the authors introduce a new kind of approximations of the identity with exponential decay (for short, exp-ATI). Via such an exp-ATI, motivated by another creative idea of Y. Han et al. to merge the aforementioned orthonormal bases of regular wavelets into the frame of the existed distributional theory on spaces of homogeneous type, the authors establish the homogeneous continuous/discrete Calderón reproducing formulae on (X, d, µ), as well as their inhomogeneous counterparts. The novelty of this article exists in that d is only assumed to be a quasi-metric and the underlying measure µ a doubling measure, not necessary to satisfy the reverse doubling condition. It is well known that Calderón reproducing formulae are the cornerstone to develop analysis and, especially, harmonic analysis on spaces of homogeneous type.
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