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Variable weak Hardy spaces WHLp(·)(ℝn) associated with operators satisfying Davies–Gaffney estimates

  • Ciqiang Zhuo and Dachun Yang ORCID logo EMAIL logo
Published/Copyright: December 5, 2018
Forum Mathematicum
From the journal Forum Mathematicum Volume 31 Issue 3

Abstract

Let p⁢(⋅):ℝn→[0,1] be a variable exponent function satisfying the globally log-Hölder continuous condition, and L a one-to-one operator of type ω in L2⁢(ℝn), with ω∈[0,π/2), which has a bounded holomorphic functional calculus and satisfies the Davies–Gaffney estimates. In this article, we introduce the variable weak Hardy space WHLp⁢(⋅)⁢(ℝn), associated with L via the corresponding square function. Its molecular characterization is then established by means of the atomic decomposition of the variable weak tent space WTp⁢(⋅)⁢(ℝ+n+1), which is also obtained in this article. In particular, when L is non-negative and self-adjoint, we obtain the atomic characterization of WHLp⁢(⋅)⁢(ℝn). As an application of the molecular characterization, when L is the second-order divergence form elliptic operator with complex bounded measurable coefficients, we prove that the associated Riesz transform ∇⁡L-1/2 is bounded from WHLp⁢(⋅)⁢(ℝn) to the variable weak Hardy space WHp⁢(⋅)⁢(ℝn). Moreover, when L is non-negative and self-adjoint with the kernels of {e-t⁢L}t>0 satisfying the Gaussian upper bound estimates, the atomic characterization of WHLp⁢(⋅)⁢(ℝn) is further used to characterize this space via non-tangential maximal functions.

Keywords: Weak Hardy space; variable exponent; operator; Davies–Gaffney estimate; atom; molecule; maximal function
MSC 2010: 42B30; 42B35; 42B25

Communicated by Christopher D. Sogge

Funding source: National Natural Science Foundation of China

Award Identifier / Grant number: 11701174

Award Identifier / Grant number: 11726621

Award Identifier / Grant number: 11571039

Award Identifier / Grant number: 11761131002

Award Identifier / Grant number: 11671185

Funding source: Education Department of Hunan Province

Award Identifier / Grant number: 17B159

Funding statement: The first author is supported by the Construct Program of the Key Discipline in Hunan Province, the Scientific Research Fund of Hunan Provincial Education Department (Grant No. 17B159) and Hunan Natural Science Foundation (Grant No: 2018JJ3321). This project is also supported by the National Natural Science Foundation of China (Grant Nos. 11701174, 11726621, 11571039, 11761131002 and 11671185).

Acknowledgements

The authors would like to express their deep thanks to the referee for his very careful reading and several useful comments which improve the presentation of this article.

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Received: 2018-05-20
Revised: 2018-09-27
Published Online: 2018-12-05
Published in Print: 2019-05-01

© 2019 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. Simplicity of skew inverse semigroup rings with applications to Steinberg algebras and topological dynamics
  3. Locally conformal symplectic structures on Lie algebras of type I and their solvmanifolds
  4. Variable weak Hardy spaces WHLp(·)(ℝn) associated with operators satisfying Davies–Gaffney estimates
  5. Sublinear operators on weighted Hardy spaces with variable exponents
  6. Hereditarily minimal topological groups
  7. Rational torsion of generalized Jacobians of modular and Drinfeld modular curves
  8. Morita homotopy theory for (∞,1)-categories and ∞-operads
  9. Homomorphisms into totally disconnected, locally compact groups with dense image
  10. On the Ramanujan–Petersson conjecture for modular forms of half-integral weight
  11. Uniform continuity of nonautonomous superposition operators in ΛBV-spaces
  12. Simple modules over the Lie algebras of divergence zero vector fields on a torus
  13. Effective bounds for the Andrews spt-function
  14. Free subgroups in k(x1,... ,xn)(X;σ) and k(x,y)(k;σ)
  15. A geometric proof of an equivariant Pieri rule for flag manifolds
  16. On relations between weak and strong type inequalities for modified maximal operators on non-doubling metric measure spaces
  17. Global well-posedness and temporal decay estimates to the fractional Cahn–Hilliard equation in ℝN
https://doi.org/10.1515/forum-2018-0125
Keywords for this article
Weak Hardy space; variable exponent; operator; Davies–Gaffney estimate; atom; molecule; maximal function

Articles in the same Issue

  1. Frontmatter
  2. Simplicity of skew inverse semigroup rings with applications to Steinberg algebras and topological dynamics
  3. Locally conformal symplectic structures on Lie algebras of type I and their solvmanifolds
  4. Variable weak Hardy spaces WHLp(·)(ℝn) associated with operators satisfying Davies–Gaffney estimates
  5. Sublinear operators on weighted Hardy spaces with variable exponents
  6. Hereditarily minimal topological groups
  7. Rational torsion of generalized Jacobians of modular and Drinfeld modular curves
  8. Morita homotopy theory for (∞,1)-categories and ∞-operads
  9. Homomorphisms into totally disconnected, locally compact groups with dense image
  10. On the Ramanujan–Petersson conjecture for modular forms of half-integral weight
  11. Uniform continuity of nonautonomous superposition operators in ΛBV-spaces
  12. Simple modules over the Lie algebras of divergence zero vector fields on a torus
  13. Effective bounds for the Andrews spt-function
  14. Free subgroups in k(x1,... ,xn)(X;σ) and k(x,y)(k;σ)
  15. A geometric proof of an equivariant Pieri rule for flag manifolds
  16. On relations between weak and strong type inequalities for modified maximal operators on non-doubling metric measure spaces
  17. Global well-posedness and temporal decay estimates to the fractional Cahn–Hilliard equation in ℝN
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