We say that a function space Z is factorable by X when there exists a third function space Y such that each f from Z admits factorization f = gh, where g, h belong to X, Y, respectively, and parall ...
{"title":"Regularization for Lozanovskii's type factorization with applications","authors":"Karol Leśnik, L. Maligranda, P. Mleczko","doi":"10.5186/aasfm.2020.4545","DOIUrl":"https://doi.org/10.5186/aasfm.2020.4545","url":null,"abstract":"We say that a function space Z is factorable by X when there exists a third function space Y such that each f from Z admits factorization f = gh, where g, h belong to X, Y, respectively, and parall ...","PeriodicalId":50787,"journal":{"name":"Annales Academiae Scientiarum Fennicae-Mathematica","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2020-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72457848","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We characterize the boundedness of Hankel bilinear forms on a product of generalized Fock–Sobolev spaces on C with respect to the weight (1 + |z|)e α 2 |z| , for l ≥ 1, α > 0 and ρ ∈ R. We obtain a weak decomposition of the Bergman kernel with estimates and a Littlewood– Paley formula, which are key ingredients in the proof of our main results. As an application, we characterize the boundedness, compactness and the membership in the Schatten class of small Hankel operators on these spaces.
{"title":"Hankel bilinear forms on generalized Fock–Sobolev spaces on C^n","authors":"C. Cascante, J. Fàbrega, D. Pascuas","doi":"10.5186/aasfm.2020.4546","DOIUrl":"https://doi.org/10.5186/aasfm.2020.4546","url":null,"abstract":"We characterize the boundedness of Hankel bilinear forms on a product of generalized Fock–Sobolev spaces on C with respect to the weight (1 + |z|)e α 2 |z| , for l ≥ 1, α > 0 and ρ ∈ R. We obtain a weak decomposition of the Bergman kernel with estimates and a Littlewood– Paley formula, which are key ingredients in the proof of our main results. As an application, we characterize the boundedness, compactness and the membership in the Schatten class of small Hankel operators on these spaces.","PeriodicalId":50787,"journal":{"name":"Annales Academiae Scientiarum Fennicae-Mathematica","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2020-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82555292","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
First, we show that a projective measured foliation is a Busemann point, in Gardiner–Masur boundary, if and only if it is indecomposable. Let f : Tg,n → Tg,n be a totally geodesic homeomorphism and suppose that f admits a homeomorphic extension to ∂GMTg,n. We show that f induces a simplicial automorphism of curve complex. Moreover, the restriction of f on Tg,n is an isometry. As an application, we obtain an alternative proof of Royden’s Theorem.
{"title":"Totally geodesic homeomorphisms between Teichmüller spaces","authors":"D. Tan","doi":"10.5186/aasfm.2020.4538","DOIUrl":"https://doi.org/10.5186/aasfm.2020.4538","url":null,"abstract":"First, we show that a projective measured foliation is a Busemann point, in Gardiner–Masur boundary, if and only if it is indecomposable. Let f : Tg,n → Tg,n be a totally geodesic homeomorphism and suppose that f admits a homeomorphic extension to ∂GMTg,n. We show that f induces a simplicial automorphism of curve complex. Moreover, the restriction of f on Tg,n is an isometry. As an application, we obtain an alternative proof of Royden’s Theorem.","PeriodicalId":50787,"journal":{"name":"Annales Academiae Scientiarum Fennicae-Mathematica","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2020-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87803293","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
. Let B σ 2 , ∞ , B σ 2 , 2 denote the Besov spaces defined on a compact set K ⊂ R d that is equipped with an α -regular measure µ ( K is called an α -set). The critical exponent σ ∗ is the supremum of the σ such that B σ 2 , 2 ∩ C ( K ) is dense in C ( K ) . It is known that B σ 2 , 2 is the domain of a non-local regular Dirichlet form, and for certain standard self-similar set, B σ ∗ 2 , ∞ is the domain of a local regular Dirichlet form. In this paper, we study, on the homogenous p.c.f. self-similar sets (which are α -sets), the convergence of the B σ 2 , 2 -norm to the B σ ∗ 2 , ∞ -norm as σ (cid:37) σ ∗ and the associated Dirichlet forms. The theorem extends a celebrate result of Bourgain, Brezis and Mironescu [4] on Euclidean domains, and the more recent results on some self-similar sets [10, 22, 29].
{"title":"Dirichlet forms and convergence of Besov norms on self-similar sets","authors":"Qingsong Gu, K. Lau","doi":"10.5186/aasfm.2020.4536","DOIUrl":"https://doi.org/10.5186/aasfm.2020.4536","url":null,"abstract":". Let B σ 2 , ∞ , B σ 2 , 2 denote the Besov spaces defined on a compact set K ⊂ R d that is equipped with an α -regular measure µ ( K is called an α -set). The critical exponent σ ∗ is the supremum of the σ such that B σ 2 , 2 ∩ C ( K ) is dense in C ( K ) . It is known that B σ 2 , 2 is the domain of a non-local regular Dirichlet form, and for certain standard self-similar set, B σ ∗ 2 , ∞ is the domain of a local regular Dirichlet form. In this paper, we study, on the homogenous p.c.f. self-similar sets (which are α -sets), the convergence of the B σ 2 , 2 -norm to the B σ ∗ 2 , ∞ -norm as σ (cid:37) σ ∗ and the associated Dirichlet forms. The theorem extends a celebrate result of Bourgain, Brezis and Mironescu [4] on Euclidean domains, and the more recent results on some self-similar sets [10, 22, 29].","PeriodicalId":50787,"journal":{"name":"Annales Academiae Scientiarum Fennicae-Mathematica","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2020-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79498133","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Weak estimates for the maximal and Riesz potential operators on non-homogeneous central Morrey type spaces in L^1 over metric measure spaces","authors":"Katsuo Matsuoka, Y. Mizuta, T. Shimomura","doi":"10.5186/aasfm.2020.4561","DOIUrl":"https://doi.org/10.5186/aasfm.2020.4561","url":null,"abstract":"","PeriodicalId":50787,"journal":{"name":"Annales Academiae Scientiarum Fennicae-Mathematica","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2020-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78381513","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
where dA(z) = dx dy π is the normalized Lebesgue area measure on D. In this definition we understand that the sum does not exist if n = 0. Throughout this paper ω satisfies ω̂(z) = ́ 1 |z| ω(s) ds > 0 for all z ∈ D, for otherwise Aω,n = H(D). We write A p ω = A p ω,0 and D ω = A p ω,1 for the weighted Bergman and Dirichlet spaces, respectively. As usual, Aα and D p α denote the classical weighted Bergman and Dirichlet spaces induced by the standard radial weight ω(z) = (1−|z|), where −1 < α < ∞. For f ∈ H(D) and 0 < r < 1, set
其中dA(z) = dx dy π是d上的标准化勒贝格面积度量。在这个定义中，我们理解如果n = 0，则和不存在。在本文中，对于所有z∈D， ω满足ω ω(z) = 1 |z| ω(s) ds > 0，否则为Aω，n = H(D)。对于加权的Bergman和Dirichlet空间，我们分别写成A p ω = A p ω，0和D ω = A p ω，1。通常，Aα和D p α表示由标准径向权ω(z) =(1−|z|)导出的经典加权Bergman和Dirichlet空间，其中−1 < α <∞。对于f∈H(D)且0 < r < 1，设
{"title":"Closure of Bergman and Dirichlet spaces in the Bloch norm","authors":"Bin Liu, J. Rättyä","doi":"10.5186/aasfm.2020.4533","DOIUrl":"https://doi.org/10.5186/aasfm.2020.4533","url":null,"abstract":"where dA(z) = dx dy π is the normalized Lebesgue area measure on D. In this definition we understand that the sum does not exist if n = 0. Throughout this paper ω satisfies ω̂(z) = ́ 1 |z| ω(s) ds > 0 for all z ∈ D, for otherwise Aω,n = H(D). We write A p ω = A p ω,0 and D ω = A p ω,1 for the weighted Bergman and Dirichlet spaces, respectively. As usual, Aα and D p α denote the classical weighted Bergman and Dirichlet spaces induced by the standard radial weight ω(z) = (1−|z|), where −1 < α < ∞. For f ∈ H(D) and 0 < r < 1, set","PeriodicalId":50787,"journal":{"name":"Annales Academiae Scientiarum Fennicae-Mathematica","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2020-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76084490","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"On A_p–A_q weighted estimates for maximal operators","authors":"A. Osȩkowski","doi":"10.5186/aasfm.2020.4544","DOIUrl":"https://doi.org/10.5186/aasfm.2020.4544","url":null,"abstract":"","PeriodicalId":50787,"journal":{"name":"Annales Academiae Scientiarum Fennicae-Mathematica","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2020-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86616290","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract. We study the averageL-dimensions of typical Borel probability measures belonging to the Gromov–Hausdorff–Prohoroff space (of all Borel probability measures with compact supports) equipped with the Gromov–Hausdorff–Prohoroff metric. Previously the lower and upper average L-dimensions of a typical measure μ have been found for q ∈ (1,∞). In this paper we determine the lower and upper average L-dimensions of a typical measure μ in the two limiting cases: q = 1 and q = ∞. In particular, we prove that a typical measure μ is as irregular as possible: for q = 1 and q = ∞, the lower average L-dimension attains the smallest possible value, namely 0, and the upper average L-dimension attains the largest possible value, namely ∞. The proofs rely on some non-trivial semi-continuity properties of L-dimensions that may be of interest in their own right.
{"title":"On the average L^q-dimensions of typical measures belonging to the Gromov–Hausdorff–Prohoroff space. The limiting cases: q = 1 and q = ∞","authors":"L. Olsen","doi":"10.5186/aasfm.2020.4535","DOIUrl":"https://doi.org/10.5186/aasfm.2020.4535","url":null,"abstract":"Abstract. We study the averageL-dimensions of typical Borel probability measures belonging to the Gromov–Hausdorff–Prohoroff space (of all Borel probability measures with compact supports) equipped with the Gromov–Hausdorff–Prohoroff metric. Previously the lower and upper average L-dimensions of a typical measure μ have been found for q ∈ (1,∞). In this paper we determine the lower and upper average L-dimensions of a typical measure μ in the two limiting cases: q = 1 and q = ∞. In particular, we prove that a typical measure μ is as irregular as possible: for q = 1 and q = ∞, the lower average L-dimension attains the smallest possible value, namely 0, and the upper average L-dimension attains the largest possible value, namely ∞. The proofs rely on some non-trivial semi-continuity properties of L-dimensions that may be of interest in their own right.","PeriodicalId":50787,"journal":{"name":"Annales Academiae Scientiarum Fennicae-Mathematica","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2020-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79586509","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"A note on Lusin's condition (N) for W_loc^1,n-mappings with convex potentials","authors":"Diego Maldonado","doi":"10.5186/aasfm.2020.4555","DOIUrl":"https://doi.org/10.5186/aasfm.2020.4555","url":null,"abstract":"","PeriodicalId":50787,"journal":{"name":"Annales Academiae Scientiarum Fennicae-Mathematica","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2020-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85318571","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
V. Gutlyanskiĭ, V. Ryazanov, E. Yakubov, A. Yefimushkin
We study the Hilbert boundary value problem for the Beltrami equation in the Jordan domains satisfying the quasihyperbolic boundary condition by Gehring–Martio, generally speaking, without (A)-condition by Ladyzhenskaya–Ural’tseva that was standard for boundary value problems in the PDE theory. Assuming that the coefficients of the problem are functions of countable bounded variation and the boundary data are measurable with respect to the logarithmic capacity, we prove the existence of the generalized regular solutions. As a consequence, we derive the existence of nonclassical solutions of the Dirichlet, Neumann and Poincaré boundary value problems for generalizations of the Laplace equation in anisotropic and inhomogeneous media.
{"title":"On Hilbert boundary value problem for Beltrami equation","authors":"V. Gutlyanskiĭ, V. Ryazanov, E. Yakubov, A. Yefimushkin","doi":"10.5186/aasfm.2020.4552","DOIUrl":"https://doi.org/10.5186/aasfm.2020.4552","url":null,"abstract":"We study the Hilbert boundary value problem for the Beltrami equation in the Jordan domains satisfying the quasihyperbolic boundary condition by Gehring–Martio, generally speaking, without (A)-condition by Ladyzhenskaya–Ural’tseva that was standard for boundary value problems in the PDE theory. Assuming that the coefficients of the problem are functions of countable bounded variation and the boundary data are measurable with respect to the logarithmic capacity, we prove the existence of the generalized regular solutions. As a consequence, we derive the existence of nonclassical solutions of the Dirichlet, Neumann and Poincaré boundary value problems for generalizations of the Laplace equation in anisotropic and inhomogeneous media.","PeriodicalId":50787,"journal":{"name":"Annales Academiae Scientiarum Fennicae-Mathematica","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2020-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78085850","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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