scholarly journals Exponential-Square Integrability, Weighted Inequalities for the Square Functions Associated to Operators, and Applications

2020 ◽  
Author(s):  
Peng Chen ◽  
Xuan Thinh Duong ◽  
Liangchuan Wu ◽  
Lixin Yan
Keyword(s):  
Analytic Semigroup ◽  
Adjoint Operator ◽  
Square Function ◽  
Weighted Norm ◽  
Square Functions ◽  
Spectral Multipliers ◽  
Eigenvalue Estimates ◽  
Preservation Condition ◽  
Square Integrability ◽  
The Laplace Operator

Abstract Let $X$ be a metric space with a doubling measure. Let $L$ be a nonnegative self-adjoint operator acting on $L^2(X)$, hence $L$ generates an analytic semigroup $e^{-tL}$. Assume that the kernels $p_t(x,y)$ of $e^{-tL}$ satisfy Gaussian upper bounds and Hölder continuity in $x$, but we do not require the semigroup to satisfy the preservation condition $e^{-tL}1 = 1$. In this article we aim to establish the exponential-square integrability of a function whose square function associated to an operator $L$ is bounded, and the proof is new even for the Laplace operator on the Euclidean spaces ${\mathbb R^n}$. We then apply this result to obtain: (1) estimates of the norm on $L^p$ as $p$ becomes large for operators such as the square functions or spectral multipliers; (2) weighted norm inequalities for the square functions; and (3) eigenvalue estimates for Schrödinger operators on ${\mathbb R}^n$ or Lipschitz domains of ${\mathbb R}^n$.

scholarly journals Notes on boundedness of spectral multipliers on Hardy spaces associated to operators

2011 ◽  
Vol 203 ◽  
pp. 109-122
Author(s):  
Bui The Anh
Keyword(s):  
Hardy Space ◽  
Hardy Spaces ◽  
Analytic Semigroup ◽  
Adjoint Operator ◽  
Upper Bounds ◽  
Space Of Homogeneous Type ◽  
Homogeneous Type ◽  
Spectral Multipliers ◽  
Spectral Multiplier ◽  
Suitable Function

AbstractLetLbe a nonnegative self-adjoint operator onL2(X), whereXis a space of homogeneous type. Assume thatLgenerates an analytic semigroupe–tlwhose kernel satisfies the standard Gaussian upper bounds. We prove that the spectral multiplierF(L) is bounded onfor 0< p< 1, the Hardy space associated to operatorL, whenFis a suitable function.

scholarly journals Notes on boundedness of spectral multipliers on Hardy spaces associated to operators

2011 ◽  
Vol 203 ◽  
pp. 109-122 ◽  
Author(s):  
Bui The Anh
Keyword(s):  
Hardy Space ◽  
Hardy Spaces ◽  
Analytic Semigroup ◽  
Adjoint Operator ◽  
Upper Bounds ◽  
Space Of Homogeneous Type ◽  
Homogeneous Type ◽  
Spectral Multipliers ◽  
Spectral Multiplier ◽  
Suitable Function

AbstractLet L be a nonnegative self-adjoint operator on L2 (X), where X is a space of homogeneous type. Assume that L generates an analytic semigroup e–tl whose kernel satisfies the standard Gaussian upper bounds. We prove that the spectral multiplier F(L) is bounded on for 0 < p < 1, the Hardy space associated to operator L, when F is a suitable function.

scholarly journals A note on Berezin-Toeplitz quantization of the Laplace operator

2015 ◽  
Vol 2 (1) ◽  
Author(s):  
Alberto Della Vedova
Keyword(s):  
Line Bundle ◽  
Laplace Operator ◽  
Adjoint Operator ◽  
Holomorphic Sections ◽  
The Laplace Operator ◽  
Hodge Manifold

AbstractGiven a Hodge manifold, it is introduced a self-adjoint operator on the space of endomorphisms of the global holomorphic sections of the polarization line bundle. Such operator is shown to approximate the Laplace operator on functions when composed with Berezin-Toeplitz quantization map and its adjoint, up to an error which tends to zero when taking higher powers of the polarization line bundle.

scholarly journals Multilinear Square Functions with Kernels of Dini’s Type

2016 ◽  
Vol 2016 ◽  
pp. 1-11
Author(s):  
Zengyan Si ◽  
Qingying Xue
Keyword(s):  
Weak Type ◽  
Square Function ◽  
Square Functions ◽  
Type Spaces

LetTbe a multilinear square function with a kernel satisfying Dini(1) condition and letT⁎be the corresponding multilinear maximal square function. In this paper, first, we showed thatTis bounded fromL1×⋯×L1toL1/m,∞.Secondly, we obtained that if eachpi>1, thenTandT⁎are bounded fromLp1(ω1)×⋯×Lpm(ωm)toLp(νω→)and if there ispi=1, thenTandT⁎are bounded fromLp1(ω1)×⋯×Lpm(ωm)toLp,∞(νω→), whereνω→=∏i=1mωip/pi.Furthermore, we established the weighted strong and weak type boundedness forTandT⁎on weighted Morrey type spaces, respectively.

Spectral Multipliers for Laplacians Associated with some Dirichlet Forms

2008 ◽  
Vol 51 (3) ◽  
pp. 581-607 ◽  
Author(s):  
Andrea Carbonaro ◽  
Giancarlo Mauceri ◽  
Stefano Meda
Keyword(s):  
Lebesgue Measure ◽  
Essential Spectrum ◽  
Dirichlet Form ◽  
Adjoint Operator ◽  
Dirichlet Forms ◽  
The Self ◽  
Spectral Multipliers ◽  
Curvature Condition ◽  
Image Position ◽  
Conditions At Infinity

AbstractLet be the self-adjoint operator associated with the Dirichlet formwhere ϕ is a positive C2 function, dλϕ = ϕdλ and λ denotes Lebesgue measure on ℝd. We study the boundedness on Lp(λϕ) of spectral multipliers of . We prove that if ϕ grows or decays at most exponentially at infinity and satisfies a suitable ‘curvature condition’, then functions which are bounded and holomorphic in the intersection of a parabolic region and a sector and satisfy Mihlin-type conditions at infinity are spectral multipliers of Lp(λϕ). The parabolic region depends on ϕ, on p and on the infimum of the essential spectrum of the operator on L2(λϕ). The sector depends on the angle of holomorphy of the semigroup generated by on Lp(λϕ).

scholarly journals Elliptic Measures and Square Function Estimates on 1-Sided Chord-Arc Domains

2022 ◽  
Vol 32 (3) ◽  
Author(s):  
Mingming Cao ◽  
José María Martell ◽  
Andrea Olivo
Keyword(s):  
Absolute Continuity ◽  
Square Function ◽  
Surface Measure ◽  
Antisymmetric Part ◽  
Null Solution ◽  
Almost Everywhere ◽  
Square Function Estimates ◽  
The Absolute ◽  
The Laplace Operator ◽  
Conical Square Function

AbstractIn nice environments, such as Lipschitz or chord-arc domains, it is well-known that the solvability of the Dirichlet problem for an elliptic operator in $$L^p$$ L p , for some finite p, is equivalent to the fact that the associated elliptic measure belongs to the Muckenhoupt class $$A_\infty $$ A ∞ . In turn, any of these conditions occurs if and only if the gradient of every bounded null solution satisfies a Carleson measure estimate. This has been recently extended to much rougher settings such as those of 1-sided chord-arc domains, that is, sets which are quantitatively open and connected with a boundary which is Ahlfors–David regular. In this paper, we work in the same environment and consider a qualitative analog of the latter equivalence showing that one can characterize the absolute continuity of the surface measure with respect to the elliptic measure in terms of the finiteness almost everywhere of the truncated conical square function for any bounded null solution. As a consequence of our main result particularized to the Laplace operator and some previous results, we show that the boundary of the domain is rectifiable if and only if the truncated conical square function is finite almost everywhere for any bounded harmonic function. In addition, we obtain that for two given elliptic operators $$L_1$$ L 1 and $$L_2$$ L 2 , the absolute continuity of the surface measure with respect to the elliptic measure of $$L_1$$ L 1 is equivalent to the same property for $$L_2$$ L 2 provided the disagreement of the coefficients satisfy some quadratic estimate in truncated cones for almost everywhere vertex. Finally, for the case on which $$L_2$$ L 2 is either the transpose of $$L_1$$ L 1 or its symmetric part we show the equivalence of the corresponding absolute continuity upon assuming that the antisymmetric part of the coefficients has some controlled oscillation in truncated cones for almost every vertex.

scholarly journals Regularities of Time-Fractional Derivatives of Semigroups Related to Schrodinger Operators with Application to Hardy-Sobolev Spaces on Heisenberg Groups

2020 ◽  
Vol 2020 ◽  
pp. 1-21
Author(s):  
Zhiyong Wang ◽  
Chuanhong Sun ◽  
Pengtao Li
Keyword(s):  
Sobolev Spaces ◽  
Equivalence Relation ◽  
Fractional Derivatives ◽  
Sobolev Type ◽  
Square Function ◽  
Heisenberg Groups ◽  
Square Functions ◽  
Regularity Properties ◽  
Reverse Hölder ◽  
Derivatives Of

In this paper, assume that L=−Δℍn+V is a Schrödinger operator on the Heisenberg group ℍn, where the nonnegative potential V belongs to the reverse Hölder class BQ/2. By the aid of the subordinate formula, we investigate the regularity properties of the time-fractional derivatives of semigroups e−tLt>0 and e−tLt>0, respectively. As applications, using fractional square functions, we characterize the Hardy-Sobolev type space HL1,αℍn associated with L. Moreover, the fractional square function characterizations indicate an equivalence relation of two classes of Hardy-Sobolev spaces related with L.

scholarly journals Spectral Multipliers of Self-Adjoint Operators on Besov and Triebel–Lizorkin Spaces Associated to Operators

2019 ◽  
Author(s):  
The Anh Bui ◽  
Xuan Thinh Duong
Keyword(s):  
Heat Kernel ◽  
Hardy Spaces ◽  
Adjoint Operator ◽  
Space Of Homogeneous Type ◽  
Multiplier Theorem ◽  
Homogeneous Type ◽  
Spectral Multipliers ◽  
Spectral Multiplier ◽  
Gaussian Estimate ◽  
Adjoint Operators

Abstract Let $X$ be a space of homogeneous type and let $L$ be a nonnegative self-adjoint operator on $L^2(X)$ that satisfies a Gaussian estimate on its heat kernel. In this paper we prove a Hörmander-type spectral multiplier theorem for $L$ on the Besov and Triebel–Lizorkin spaces associated to $L$. Our work not only recovers the boundedness of the spectral multipliers on $L^p$ spaces and Hardy spaces associated to $L$ but also is the 1st one that proves the boundedness of a general spectral multiplier theorem on Besov and Triebel–Lizorkin spaces.

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