We show that the volume of the boundary of a bounded Sobolev $(p,q)$-extension domain is zero when $1leq q
我们证明，当 $1leq q
{"title":"The volume of the boundary of a Sobolev $(p,q)$-extension domain II","authors":"Pekka Koskela, Riddhi Mishra","doi":"arxiv-2409.01170","DOIUrl":"https://doi.org/arxiv-2409.01170","url":null,"abstract":"We show that the volume of the boundary of a bounded Sobolev\u0000$(p,q)$-extension domain is zero when $1leq q <p< frac{qn}{(n-q)}.$","PeriodicalId":501036,"journal":{"name":"arXiv - MATH - Functional Analysis","volume":"8 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142208188","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We give a direct proof of the operator valued Hardy-Littlewood maximal inequality for $2
我们直接证明了 2
{"title":"A Direct Proof of Hardy-Littlewood Maximal Inequality for Operator-valued Functions","authors":"ChianYeong Chuah, Zhenchuan Liu, Tao Mei","doi":"arxiv-2409.00752","DOIUrl":"https://doi.org/arxiv-2409.00752","url":null,"abstract":"We give a direct proof of the operator valued Hardy-Littlewood maximal\u0000inequality for $2<p<infty$.","PeriodicalId":501036,"journal":{"name":"arXiv - MATH - Functional Analysis","volume":"16 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142208185","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Xingni Jiang, Jan Harm van der Walt, Marten Wortel
We develop integration theory for integrating functions taking values into a Dedekind complete unital $f$-algebra $mathbb{L}$ with respect to $mathbb{L}$-valued measures. We then discuss and prove completeness results of $mathbb{L}$-valued $L^p$-spaces.
{"title":"L-valued integration","authors":"Xingni Jiang, Jan Harm van der Walt, Marten Wortel","doi":"arxiv-2408.17306","DOIUrl":"https://doi.org/arxiv-2408.17306","url":null,"abstract":"We develop integration theory for integrating functions taking values into a\u0000Dedekind complete unital $f$-algebra $mathbb{L}$ with respect to\u0000$mathbb{L}$-valued measures. We then discuss and prove completeness results of\u0000$mathbb{L}$-valued $L^p$-spaces.","PeriodicalId":501036,"journal":{"name":"arXiv - MATH - Functional Analysis","volume":"6 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142208191","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
For every $alpha in (0,+infty)$ and $p,q in (1,+infty)$ let $T_alpha$ be the operator $L^p[0,1]to L^q[0,1]$ defined via the equality $(T_alpha f)(x) := int_0^{x^alpha} f(y) d y$. We study the norms of $T_alpha$ for every $p$, $q$. In the case $p=q$ we further study its spectrum, point spectrum, eigenfunctions, and the norms of its iterates. Moreover, for the case $p=q=2$ we determine the point spectrum and eigenfunctions for $T^*_alpha T_alpha$, where $T^*_alpha$ is the adjoint operator.
{"title":"A one parameter family of Volterra-type operators","authors":"Francesco Battistoni, Giuseppe Molteni","doi":"arxiv-2408.17124","DOIUrl":"https://doi.org/arxiv-2408.17124","url":null,"abstract":"For every $alpha in (0,+infty)$ and $p,q in (1,+infty)$ let $T_alpha$\u0000be the operator $L^p[0,1]to L^q[0,1]$ defined via the equality $(T_alpha\u0000f)(x) := int_0^{x^alpha} f(y) d y$. We study the norms of $T_alpha$ for\u0000every $p$, $q$. In the case $p=q$ we further study its spectrum, point\u0000spectrum, eigenfunctions, and the norms of its iterates. Moreover, for the case\u0000$p=q=2$ we determine the point spectrum and eigenfunctions for $T^*_alpha\u0000T_alpha$, where $T^*_alpha$ is the adjoint operator.","PeriodicalId":501036,"journal":{"name":"arXiv - MATH - Functional Analysis","volume":"314 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142208192","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let $A$ be an $mtimes m$ complex matrix and let $lambda _1, lambda _2, ldots , lambda _m$ be the eigenvalues of $A$ arranged such that $|lambda _1|geq |lambda _2|geq cdots geq |lambda _m|$ and for $ngeq 1,$ let $s^{(n)}_1geq s^{(n)}_2geq cdots geq s^{(n)}_m$ be the singular values of $A^n$. Then a famous theorem of Yamamoto (1967) states that $$lim _{nto infty}(s^{(n)}_j )^{frac{1}{n}}= |lambda _j|, ~~forall ,1leq jleq m.$$ Recently S. Nayak strengthened this result very significantly by showing that the sequence of matrices $|A^n|^{frac{1}{n}}$ itself converges to a positive matrix $B$ whose eigenvalues are $|lambda _1|,|lambda _2|,$ $ldots , |lambda _m|.$ Here this theorem has been extended to arbitrary compact operators on infinite dimensional complex separable Hilbert spaces. The proof makes use of Nayak's theorem, Stone-Weirstrass theorem, Borel-Caratheodory theorem and some technical results of Anselone and Palmer on collectively compact operators. Simple examples show that the result does not hold for general bounded operators.
{"title":"Nayak's theorem for compact operators","authors":"B V Rajarama Bhat, Neeru Bala","doi":"arxiv-2408.16994","DOIUrl":"https://doi.org/arxiv-2408.16994","url":null,"abstract":"Let $A$ be an $mtimes m$ complex matrix and let $lambda _1, lambda _2,\u0000ldots , lambda _m$ be the eigenvalues of $A$ arranged such that $|lambda\u0000_1|geq |lambda _2|geq cdots geq |lambda _m|$ and for $ngeq 1,$ let\u0000$s^{(n)}_1geq s^{(n)}_2geq cdots geq s^{(n)}_m$ be the singular values of\u0000$A^n$. Then a famous theorem of Yamamoto (1967) states that $$lim _{nto\u0000infty}(s^{(n)}_j )^{frac{1}{n}}= |lambda _j|, ~~forall ,1leq jleq m.$$\u0000Recently S. Nayak strengthened this result very significantly by showing that\u0000the sequence of matrices $|A^n|^{frac{1}{n}}$ itself converges to a positive\u0000matrix $B$ whose eigenvalues are $|lambda _1|,|lambda _2|,$ $ldots ,\u0000|lambda _m|.$ Here this theorem has been extended to arbitrary compact\u0000operators on infinite dimensional complex separable Hilbert spaces. The proof\u0000makes use of Nayak's theorem, Stone-Weirstrass theorem, Borel-Caratheodory\u0000theorem and some technical results of Anselone and Palmer on collectively\u0000compact operators. Simple examples show that the result does not hold for\u0000general bounded operators.","PeriodicalId":501036,"journal":{"name":"arXiv - MATH - Functional Analysis","volume":"14 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142208193","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this work, we study three problems related to the $L_1$-influence on quantum Boolean cubes. In the first place, we obtain a dimension free bound for $L_1$-influence, which implies the quantum $L^1$-KKL Theorem result obtained by Rouze, Wirth and Zhang. Beyond that, we also obtain a high order quantum Talagrand inequality and quantum $L^1$-KKL theorem. Lastly, we prove a quantitative relation between the noise stability and $L^1$-influence. To this end, our technique involves the random restrictions method as well as semigroup theory.
{"title":"Geometric influences on quantum Boolean cubes","authors":"David P. Blecher, Li Gao, Bang Xu","doi":"arxiv-2409.00224","DOIUrl":"https://doi.org/arxiv-2409.00224","url":null,"abstract":"In this work, we study three problems related to the $L_1$-influence on\u0000quantum Boolean cubes. In the first place, we obtain a dimension free bound for\u0000$L_1$-influence, which implies the quantum $L^1$-KKL Theorem result obtained by\u0000Rouze, Wirth and Zhang. Beyond that, we also obtain a high order quantum\u0000Talagrand inequality and quantum $L^1$-KKL theorem. Lastly, we prove a\u0000quantitative relation between the noise stability and $L^1$-influence. To this\u0000end, our technique involves the random restrictions method as well as semigroup\u0000theory.","PeriodicalId":501036,"journal":{"name":"arXiv - MATH - Functional Analysis","volume":"25 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142208187","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper consists of two parts. In the first half, we solve the question raised by Heil as to whether the atom of a Gabor frame must be in $M^p(mathbb{R})$ for some $1
本文由两部分组成。在前半部分，我们解决了海尔（Heil）提出的问题，即在某些$1
{"title":"Gabor frames with atoms in M^q(R) but not in M^p(R) for any 1leq p < q leq 2","authors":"Pu-Ting Yu","doi":"arxiv-2408.16593","DOIUrl":"https://doi.org/arxiv-2408.16593","url":null,"abstract":"This paper consists of two parts. In the first half, we solve the question\u0000raised by Heil as to whether the atom of a Gabor frame must be in\u0000$M^p(mathbb{R})$ for some $1<p<2$. Specifically, for each $0<alpha beta leq 1$ and $1<qleq 2$ we explicitly construct\u0000Gabor frames $mathcal{G}(g,alpha,beta)$ with atoms in $M^q(mathbb{R})$ but\u0000not in $M^{p}(mathbb{R})$ for any $1leq p<q$. To construct such Gabor frames,\u0000we use box functions as the window functions and show that $$f = sum_{k,nin\u0000mathbb{Z}} langle f,M_{beta n}T_{alpha k}\u0000mathcal{F}(chi_{[0,alpha]})rangle M_{beta n}T_{alpha k} (\u0000mathcal{F}(chi_{[0,alpha]}))$$ holds for $fin M^{p,q}(mathbb{R})$ with\u0000unconditional convergence of the series for any $0<alphabeta leq 1$,\u0000$1<p<infty$ and $1leq q<infty$. In the second half of this paper, we study two questions related to\u0000unconditional convergence of Gabor expansions in modulation spaces. Under the\u0000assumption that the window functions are chosen from $M^p(mathbb{R})$ for some\u0000$1leq pleq 2,$ we will prove several equivalent statements that the equation\u0000$f = sum_{k,nin mathbb{Z}} langle f, M_{beta n}T_{alpha k} gamma rangle\u0000M_{beta n}T_{alpha k} g$ can be extended from $L^2(mathbb{R})$ to\u0000$M^q(mathbb{R})$ for all $fin M^q(mathbb{R})$ and all $pleq qleq p'$ with\u0000unconditional convergence of the series. Finally, we characterize all Gabor\u0000systems ${M_{beta n}T_{alpha k}g}_{n,kin mathbb{Z}}$ in\u0000$M^{p,q}(mathbb{R})$ for any $1leq p,q<infty$ for which $f = sum langle f,\u0000gamma_{k,n} rangle M_{beta n}T_{alpha k} g$ with unconditional convergence\u0000of the series for all $f$ in $M^{p,q}(mathbb{R})$ and all alternative duals\u0000${gamma_{k,n}}_{k,nin mathbb{Z}}$ of ${M_{beta n}T_{alpha k}\u0000g}_{n,kin mathbb{Z}}$.","PeriodicalId":501036,"journal":{"name":"arXiv - MATH - Functional Analysis","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142208195","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We obtain asymptotically sharp identification of fractional Sobolev spaces $ W^{s}_{p,q}$, extension spaces $E^{s}_{p,q}$, and Triebel-Lizorkin spaces $dot{F}^s_{p,q}$. In particular we obtain for $W^{s}_{p,q}$ and $E^{s}_{p,q}$ a stability theory a la Bourgain-Brezis-Mironescu as $s to 1$, answering a question raised by Brazke--Schikorra--Yung. Part of the results are new even for $p=q$.
{"title":"Asymptotic Behaviour of three fractional spaces","authors":"Ahmed Dughayshim","doi":"arxiv-2408.16894","DOIUrl":"https://doi.org/arxiv-2408.16894","url":null,"abstract":"We obtain asymptotically sharp identification of fractional Sobolev spaces $\u0000W^{s}_{p,q}$, extension spaces $E^{s}_{p,q}$, and Triebel-Lizorkin spaces\u0000$dot{F}^s_{p,q}$. In particular we obtain for $W^{s}_{p,q}$ and $E^{s}_{p,q}$\u0000a stability theory a la Bourgain-Brezis-Mironescu as $s to 1$, answering a\u0000question raised by Brazke--Schikorra--Yung. Part of the results are new even\u0000for $p=q$.","PeriodicalId":501036,"journal":{"name":"arXiv - MATH - Functional Analysis","volume":"27 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142208194","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
For non-empty sets X we define notions of distance and pseudo metric with values in a partially ordered set that has a smallest element $theta $. If $h_X$ is a distance in $X$ (respectively, a pseudo metric in $X$), then the pair $(X,h_X)$ is called a distance (respectively, a pseudo metric) space. If $(T,h_T)$ and $(X,h_X)$ are pseudo metric spaces, $(Y,h_Y)$ is a distance space, and $H(T,X)$ is a class of Lipschitz mappings $fcolon Tto X$, for a broad family of mappings $Lambdacolon H (T,X)to Y$, we obtain a sharp inequality that estimates the deviation $h_Y(Lambda f(cdot),Lambda f(t))$ in terms of the function $h_T(cdot, t)$. We also show that many known estimates of such kind are contained in our general result.
{"title":"Ostrowski-type inequalities in abstract distance spaces","authors":"Vladyslav Babenko, Vira Babenko, Oleg Kovalenko","doi":"arxiv-2408.15579","DOIUrl":"https://doi.org/arxiv-2408.15579","url":null,"abstract":"For non-empty sets X we define notions of distance and pseudo metric with\u0000values in a partially ordered set that has a smallest element $theta $. If\u0000$h_X$ is a distance in $X$ (respectively, a pseudo metric in $X$), then the\u0000pair $(X,h_X)$ is called a distance (respectively, a pseudo metric) space. If\u0000$(T,h_T)$ and $(X,h_X)$ are pseudo metric spaces, $(Y,h_Y)$ is a distance\u0000space, and $H(T,X)$ is a class of Lipschitz mappings $fcolon Tto X$, for a\u0000broad family of mappings $Lambdacolon H (T,X)to Y$, we obtain a sharp\u0000inequality that estimates the deviation $h_Y(Lambda f(cdot),Lambda f(t))$ in\u0000terms of the function $h_T(cdot, t)$. We also show that many known estimates\u0000of such kind are contained in our general result.","PeriodicalId":501036,"journal":{"name":"arXiv - MATH - Functional Analysis","volume":"8 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142208198","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we first give a sufficiently condition for precompactness in the matrix-weighted Lebesgue spaces with variable exponent by translation operator. Then we obtain a criterion for precompactness in the matrix-weighted Lebesgue space with variable exponent by average operator. Next, we give a criterion for precompactness in the matrix-weighted Lebesgue space with variable exponent by approximate identity. Finally, precompactness in the matrix-weighted Sobolev space with variable exponent is also considered.
{"title":"Precompact Sets in Matrix Weighted Lebesgue Spaces with Variable Exponent","authors":"Shengrong Wang, Pengfei Guo, Jingshi Xu","doi":"arxiv-2408.15599","DOIUrl":"https://doi.org/arxiv-2408.15599","url":null,"abstract":"In this paper, we first give a sufficiently condition for precompactness in\u0000the matrix-weighted Lebesgue spaces with variable exponent by translation\u0000operator. Then we obtain a criterion for precompactness in the matrix-weighted\u0000Lebesgue space with variable exponent by average operator. Next, we give a\u0000criterion for precompactness in the matrix-weighted Lebesgue space with\u0000variable exponent by approximate identity. Finally, precompactness in the\u0000matrix-weighted Sobolev space with variable exponent is also considered.","PeriodicalId":501036,"journal":{"name":"arXiv - MATH - Functional Analysis","volume":"31 3 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142208197","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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