In this paper we prove a theorem about existence of best appraximantion in a class of spaces involving Besov spaces,via a discretization technique.It is a consequence of this theorem that rational functions and expone...In this paper we prove a theorem about existence of best appraximantion in a class of spaces involving Besov spaces,via a discretization technique.It is a consequence of this theorem that rational functions and exponencial sums are proximinal subsets of B (φ). It is also proved the proximinality of R[a,b] in B(φ) for arbitrary p,q and α.展开更多
In this paper, we show the existence and regularity of mild solutions depending on the small initial data in Besov spaces to the fractional porous medium equation. When 1 < <em>α</em> ≤ 2, we prove gl...In this paper, we show the existence and regularity of mild solutions depending on the small initial data in Besov spaces to the fractional porous medium equation. When 1 < <em>α</em> ≤ 2, we prove global well-posedness for initial data <img src="Edit_b7b43d4c-00d8-49d6-9066-97151fb5c337.bmp" alt="" /> with 1 ≤ <em>p</em> < ∞, 1 ≤ <em>q</em> ≤ ∞, and analyticity of solutions with 1 < <em>p</em> < ∞, 1 ≤ <em>q</em> ≤ ∞. In particular, we also proved that when <em>α</em> = 1, both <em>u</em> and <img src="Edit_a5af0853-8adc-4a08-b8a2-b9a70ea0f409.bmp" alt="" /> belong to <img src="Edit_03a932cc-aa58-4568-83ad-f16416cc7b71.bmp" alt="" />. We solve this equation through the contraction mapping method based on Littlewood-Paley theory and Fourier multiplier. Furthermore, we can get time decay estimates of global solutions in Besov spaces, which is <img src="Edit_083986e9-4e1c-4494-ac5d-a7d30a12df97.bmp" alt="" /> as <em>t</em> → ∞.展开更多
In this article, the author introduces a class of non-convolution Calder′on-Zygmund operators whose kernels are certain sums involving the products of Meyer wavelets and their convolutions. The boundedness on Besov s...In this article, the author introduces a class of non-convolution Calder′on-Zygmund operators whose kernels are certain sums involving the products of Meyer wavelets and their convolutions. The boundedness on Besov spaces ■p0 ,q(1 ≤p,q ≤∞) is also obtained. Moreover, as an application, the author gives a brief proof of the known result that Hrmander condition can ensure the boundedness of convolution-type Calder′on-Zygmund operators on Besov spaces ■p0 ,q(1 ≤p,q ≤∞). However, the proof is quite different from the previous one.展开更多
We study weighted holomorphic Besov spaces and their boundary values. Under certain restrictions on the weighted function and parameters, we establish the equivalent norms for holomorphic functions in terms of their b...We study weighted holomorphic Besov spaces and their boundary values. Under certain restrictions on the weighted function and parameters, we establish the equivalent norms for holomorphic functions in terms of their boundary functions. Some results about embedding and interpolation are also included.展开更多
In this paper the classical Besov spaces Bsp,q and Triebel-Lizorkin spaces Fps,q for s ∈ R are generalized in an isotropy way with the smoothness weights {|2j|lnα}∞j=0. These generalized Besov spaces and Triebel-Li...In this paper the classical Besov spaces Bsp,q and Triebel-Lizorkin spaces Fps,q for s ∈ R are generalized in an isotropy way with the smoothness weights {|2j|lnα}∞j=0. These generalized Besov spaces and Triebel-Lizorkin spaces, denoted by Bαp,q and Fpα,q for α∈ Rk and k ∈ N, respectively, keep many interesting properties, such as embedding theorems (with scales property for all smoothness weights), lifting properties for all parameters α, and duality for index 0 < p < ∞. By constructing an example, it is shown that there are infinitely many generalized Besov spaces and generalized Triebel-Lizorkin spaces lying between Bsp,q and t>s Btp,q, and between Fps,q and t>s Fpt,q, respectively.展开更多
The optimality of a density estimation on Besov spaces Bsr,q(R) for the Lp risk was established by Donoho, Johnstone, Kerkyacharian and Picard (“Density estimation by wavelet thresholding,” The Annals of Statistics,...The optimality of a density estimation on Besov spaces Bsr,q(R) for the Lp risk was established by Donoho, Johnstone, Kerkyacharian and Picard (“Density estimation by wavelet thresholding,” The Annals of Statistics, Vol. 24, No. 2, 1996, pp. 508-539.). To show the lower bound of optimal rates of convergence Rn(Bsr,q, p), they use Korostelev and Assouad lemmas. However, the conditions of those two lemmas are difficult to be verified. This paper aims to give another proof for that bound by using Fano’s Lemma, which looks a little simpler. In addition, our method can be used in many other statistical models for lower bounds of estimations.展开更多
There are two folds in this article. One fold is to characterize the Besov spaces of para-accretive type , which reduces to the classical Besov spaces when the para-accretive function is constant, by using a discrete ...There are two folds in this article. One fold is to characterize the Besov spaces of para-accretive type , which reduces to the classical Besov spaces when the para-accretive function is constant, by using a discrete Calderón-type reproducing formula and Plancherel-P?lya-type inequality associated to a para-accretive function b in Rn. The other is to show that a generalized singular integral operator T with extends to be bounded from for and , where ε is the regularity exponent of the kernel of T.展开更多
Suppose μ is a Radon measure on Rd, which may be non doubling. The only condition assumed on μ is a growth condition, namely, there is a constant Co > 0 such that for all x ∈ supp(μ) and r > 0,μ(B(x,r)) ≤ Corn...Suppose μ is a Radon measure on Rd, which may be non doubling. The only condition assumed on μ is a growth condition, namely, there is a constant Co > 0 such that for all x ∈ supp(μ) and r > 0,μ(B(x,r)) ≤ Corn, where 0 < n ≤ d. We prove T1 theorem for non doubling measures with weak kernel conditions. Our approach yields new results for kernels satisfying weakened regularity conditions, while recovering previously known Tolsa's results. We also prove T1 theorem for Besov spaces on nonhomogeneous spaces with weak kernel conditions given in [7].展开更多
In this paper,the author introduces new Triebel-Lizorkin spaces and Besov spaces associated with different homogeneities and proves that the composition of two Calderón-Zygmund singular integral operators with di...In this paper,the author introduces new Triebel-Lizorkin spaces and Besov spaces associated with different homogeneities and proves that the composition of two Calderón-Zygmund singular integral operators with different homogeneities is bounded on these new Triebel-Lizorkin spaces and Besov spaces.展开更多
In this paper,using inhomogeneous Calderon’s reproducing formulas and the space of test functions associated with a para-accretive function,the inhomogeneous Besov and TriebelLizorkin spaces are established.As applic...In this paper,using inhomogeneous Calderon’s reproducing formulas and the space of test functions associated with a para-accretive function,the inhomogeneous Besov and TriebelLizorkin spaces are established.As applications,pointwise multiplier theorems are also obtained.展开更多
We introduce the variable integral and the smooth exponent Besov spaces associated to non-negative self-adjoint operators.Then we give the equivalent norms via the Peetre type maximal functions and atomic decompositio...We introduce the variable integral and the smooth exponent Besov spaces associated to non-negative self-adjoint operators.Then we give the equivalent norms via the Peetre type maximal functions and atomic decomposition of these spaces.展开更多
In this paper we prove that the critical exponents of Besov spaces on a compact set possessing an Ahlfors regular measure is an invariant under Lipschitz transforms.Under mild conditions,the critical exponent of Besov...In this paper we prove that the critical exponents of Besov spaces on a compact set possessing an Ahlfors regular measure is an invariant under Lipschitz transforms.Under mild conditions,the critical exponent of Besov spaces of certain selfsimilar sets coincides with the walk dimension,which plays an important role in the analysis on fractals.As an application,we show examples having different critical exponents are not Lipschitz equivalent.展开更多
In this paper,we mainly investigate the Cauchy problem of the non-viscous MHD equations with magnetic diffusion.We first establish the local well-posedness(existence,uniqueness and continuous dependence)with initial d...In this paper,we mainly investigate the Cauchy problem of the non-viscous MHD equations with magnetic diffusion.We first establish the local well-posedness(existence,uniqueness and continuous dependence)with initial data(u_(0),b_(0))in critical Besov spaces B_(p,1)^(d/p+1)×B_(p,1)^(d/p)with 1≤p≤∞,and give a lifespan T of the solution which depends on the norm of the Littlewood–Paley decomposition(profile)of the initial data.Then,we prove the global existence in critical Besov spaces.In particular,the results of global existence also hold in Sobolev space C([0,∞);H~s(S~2))×(C([0,∞);H^(s-1)(S~2))∩L~2([0,∞);H~s(S~2)))with s>2,when the initial data satisfies∫_(S~2)b_(0)dx=0 and||u_(0)||B_(()∞,1~((S~2)))~1+||b_(0)||B_(()∞,1^(S~2))~0≤ε.It’s worth noting that our results imply some large and low regularity initial data for the global existence.展开更多
We establish Littlewood-Paley charaterizations of Triebel-Lizorkin spaces and Besov spaces in Euclidean spaces using several square functions defined via the spherical average,the ball average,the Bochner-Riesz means ...We establish Littlewood-Paley charaterizations of Triebel-Lizorkin spaces and Besov spaces in Euclidean spaces using several square functions defined via the spherical average,the ball average,the Bochner-Riesz means and some other well-known operators.We provide a simple proof so that we are able to extend and improve many results published in recent papers.展开更多
In this paper,the boundedness is obtained on the Triebel-Lizorkin spaces and the Besov spaces for a class of oscillatory singular integrals with Hardy kernels.
In the paper we give a trace theorem of Besov spaces Bαp,p(Rn) on a d-set.It is a kind of extension of related results of Jonsson and Wallin,which has important applications on PDE theory.
We study sufficient conditions on radial and non-radial weight functions on the upper half-plane that guarantee norm approximation of functions in weighted Bergman,weighted Dirichlet,and weighted Besov spaces on the u...We study sufficient conditions on radial and non-radial weight functions on the upper half-plane that guarantee norm approximation of functions in weighted Bergman,weighted Dirichlet,and weighted Besov spaces on the upper half-plane by dilatations and eventually by analytic polynomials.展开更多
文摘In this paper we prove a theorem about existence of best appraximantion in a class of spaces involving Besov spaces,via a discretization technique.It is a consequence of this theorem that rational functions and exponencial sums are proximinal subsets of B (φ). It is also proved the proximinality of R[a,b] in B(φ) for arbitrary p,q and α.
文摘In this paper, we show the existence and regularity of mild solutions depending on the small initial data in Besov spaces to the fractional porous medium equation. When 1 < <em>α</em> ≤ 2, we prove global well-posedness for initial data <img src="Edit_b7b43d4c-00d8-49d6-9066-97151fb5c337.bmp" alt="" /> with 1 ≤ <em>p</em> < ∞, 1 ≤ <em>q</em> ≤ ∞, and analyticity of solutions with 1 < <em>p</em> < ∞, 1 ≤ <em>q</em> ≤ ∞. In particular, we also proved that when <em>α</em> = 1, both <em>u</em> and <img src="Edit_a5af0853-8adc-4a08-b8a2-b9a70ea0f409.bmp" alt="" /> belong to <img src="Edit_03a932cc-aa58-4568-83ad-f16416cc7b71.bmp" alt="" />. We solve this equation through the contraction mapping method based on Littlewood-Paley theory and Fourier multiplier. Furthermore, we can get time decay estimates of global solutions in Besov spaces, which is <img src="Edit_083986e9-4e1c-4494-ac5d-a7d30a12df97.bmp" alt="" /> as <em>t</em> → ∞.
基金Sponsored by the NSF of South-Central University for Nationalities(YZZ08004)NNSF of China (10871209)
文摘In this article, the author introduces a class of non-convolution Calder′on-Zygmund operators whose kernels are certain sums involving the products of Meyer wavelets and their convolutions. The boundedness on Besov spaces ■p0 ,q(1 ≤p,q ≤∞) is also obtained. Moreover, as an application, the author gives a brief proof of the known result that Hrmander condition can ensure the boundedness of convolution-type Calder′on-Zygmund operators on Besov spaces ■p0 ,q(1 ≤p,q ≤∞). However, the proof is quite different from the previous one.
基金Both authors are supported in part by the Azerbaijan-U.S. Bilateral Grants Program (project ANSF Award / 3102)The second author is also supported in part by NSF grant, DMS 0200587
文摘We study weighted holomorphic Besov spaces and their boundary values. Under certain restrictions on the weighted function and parameters, we establish the equivalent norms for holomorphic functions in terms of their boundary functions. Some results about embedding and interpolation are also included.
基金Supported by NSFC of China under Grant #10571084NSC in Taipei under Grant NSC 94-2115-M-008-009(for the second author)
文摘In this paper the classical Besov spaces Bsp,q and Triebel-Lizorkin spaces Fps,q for s ∈ R are generalized in an isotropy way with the smoothness weights {|2j|lnα}∞j=0. These generalized Besov spaces and Triebel-Lizorkin spaces, denoted by Bαp,q and Fpα,q for α∈ Rk and k ∈ N, respectively, keep many interesting properties, such as embedding theorems (with scales property for all smoothness weights), lifting properties for all parameters α, and duality for index 0 < p < ∞. By constructing an example, it is shown that there are infinitely many generalized Besov spaces and generalized Triebel-Lizorkin spaces lying between Bsp,q and t>s Btp,q, and between Fps,q and t>s Fpt,q, respectively.
文摘The optimality of a density estimation on Besov spaces Bsr,q(R) for the Lp risk was established by Donoho, Johnstone, Kerkyacharian and Picard (“Density estimation by wavelet thresholding,” The Annals of Statistics, Vol. 24, No. 2, 1996, pp. 508-539.). To show the lower bound of optimal rates of convergence Rn(Bsr,q, p), they use Korostelev and Assouad lemmas. However, the conditions of those two lemmas are difficult to be verified. This paper aims to give another proof for that bound by using Fano’s Lemma, which looks a little simpler. In addition, our method can be used in many other statistical models for lower bounds of estimations.
文摘There are two folds in this article. One fold is to characterize the Besov spaces of para-accretive type , which reduces to the classical Besov spaces when the para-accretive function is constant, by using a discrete Calderón-type reproducing formula and Plancherel-P?lya-type inequality associated to a para-accretive function b in Rn. The other is to show that a generalized singular integral operator T with extends to be bounded from for and , where ε is the regularity exponent of the kernel of T.
基金The project was supported by the National Natural Science Fbundation of China(Grant No.10171111)the Foundation of Zhongshan University Advanced Research Center.
文摘Suppose μ is a Radon measure on Rd, which may be non doubling. The only condition assumed on μ is a growth condition, namely, there is a constant Co > 0 such that for all x ∈ supp(μ) and r > 0,μ(B(x,r)) ≤ Corn, where 0 < n ≤ d. We prove T1 theorem for non doubling measures with weak kernel conditions. Our approach yields new results for kernels satisfying weakened regularity conditions, while recovering previously known Tolsa's results. We also prove T1 theorem for Besov spaces on nonhomogeneous spaces with weak kernel conditions given in [7].
文摘In this paper,the author introduces new Triebel-Lizorkin spaces and Besov spaces associated with different homogeneities and proves that the composition of two Calderón-Zygmund singular integral operators with different homogeneities is bounded on these new Triebel-Lizorkin spaces and Besov spaces.
基金supported by the National Natural Science Foundation of China(11901495)Hunan Provincial NSF Project(2019JJ50573)the Scientific Research Fund of Hunan Provincial Education Department(22B0155)。
文摘In this paper,using inhomogeneous Calderon’s reproducing formulas and the space of test functions associated with a para-accretive function,the inhomogeneous Besov and TriebelLizorkin spaces are established.As applications,pointwise multiplier theorems are also obtained.
基金supported in part by the National Natural Science Foundation of China(Grant No.11761026,11761027)Natural Science Foundation of Guangxi(Grant No.2020GXNSFAA159085).
文摘We introduce the variable integral and the smooth exponent Besov spaces associated to non-negative self-adjoint operators.Then we give the equivalent norms via the Peetre type maximal functions and atomic decomposition of these spaces.
基金Foundation item: Supported by the National Natural Science Foundation of China(10771064) Supported by the Natural Science Foundation of Zhejiang Province(YT080197, Y6090036, Y6100219) Supported by the Foundation of Creative Group in Colleges and Universities of Zhejiang Province(T200924) Acknowledgement The author would like to express his thanks to his supervisor, Prof HU Zhang-jian, for his guidence.
基金The second author is supported by NSFC Nos.10631040 and 11471075。
文摘In this paper we prove that the critical exponents of Besov spaces on a compact set possessing an Ahlfors regular measure is an invariant under Lipschitz transforms.Under mild conditions,the critical exponent of Besov spaces of certain selfsimilar sets coincides with the walk dimension,which plays an important role in the analysis on fractals.As an application,we show examples having different critical exponents are not Lipschitz equivalent.
基金Supported by National Natural Science Foundation of China(Grant No.11671407 and 11701586)the Macao Science and Technology Development Fund(Grant No.0091/2018/A3)+1 种基金Guangdong Special Support Program(Grant No.8-2015)the key pro ject of NSF of Guangdong province(Grant No.2016A030311004)。
文摘In this paper,we mainly investigate the Cauchy problem of the non-viscous MHD equations with magnetic diffusion.We first establish the local well-posedness(existence,uniqueness and continuous dependence)with initial data(u_(0),b_(0))in critical Besov spaces B_(p,1)^(d/p+1)×B_(p,1)^(d/p)with 1≤p≤∞,and give a lifespan T of the solution which depends on the norm of the Littlewood–Paley decomposition(profile)of the initial data.Then,we prove the global existence in critical Besov spaces.In particular,the results of global existence also hold in Sobolev space C([0,∞);H~s(S~2))×(C([0,∞);H^(s-1)(S~2))∩L~2([0,∞);H~s(S~2)))with s>2,when the initial data satisfies∫_(S~2)b_(0)dx=0 and||u_(0)||B_(()∞,1~((S~2)))~1+||b_(0)||B_(()∞,1^(S~2))~0≤ε.It’s worth noting that our results imply some large and low regularity initial data for the global existence.
基金supported by National Natural Science Foundation of China(Grant Nos.11971295,11871108 and 11871436)Natural Science Foundation of Shanghai(No.19ZR1417600).
文摘We establish Littlewood-Paley charaterizations of Triebel-Lizorkin spaces and Besov spaces in Euclidean spaces using several square functions defined via the spherical average,the ball average,the Bochner-Riesz means and some other well-known operators.We provide a simple proof so that we are able to extend and improve many results published in recent papers.
基金Supported by the National Natural Science Foundation of China (Grant No. 11071250)
文摘In this paper,the boundedness is obtained on the Triebel-Lizorkin spaces and the Besov spaces for a class of oscillatory singular integrals with Hardy kernels.
基金Supported by the Ji’nan University’s Young Scholar’s Foundation (Grant No51208036)
文摘In the paper we give a trace theorem of Besov spaces Bαp,p(Rn) on a d-set.It is a kind of extension of related results of Jonsson and Wallin,which has important applications on PDE theory.
文摘We study sufficient conditions on radial and non-radial weight functions on the upper half-plane that guarantee norm approximation of functions in weighted Bergman,weighted Dirichlet,and weighted Besov spaces on the upper half-plane by dilatations and eventually by analytic polynomials.