Abstract
We prove a general sparse domination theorem in a space of homogeneous type, in which a vector-valued operator is controlled pointwise by a positive, local expression called a sparse operator. We use the structure of the operator to get sparse domination in which the usual ℓ1-sum in the sparse operator is replaced by an ℓr-sum. This sparse domination theorem is applicable to various operators from both harmonic analysis and (S)PDE. Using our main theorem, we prove the A2-theorem for vector-valued Calderón–Zygmund operators in a space of homogeneous type, from which we deduce an anisotropic, mixed-norm Mihlin multiplier theorem. Furthermore, we show quantitative weighted norm inequalities for the Rademacher maximal operator, for which Banach space geometry plays a major role.
Original language | English |
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Pages (from-to) | 9366-9405 |
Number of pages | 40 |
Journal | Journal of Geometric Analysis |
Volume | 31 |
Issue number | 9 |
DOIs | |
Publication status | Published - 2020 |
Keywords
- Mihlin multiplier theorem
- Muckenhoupt weight
- Rademacher maximal operator
- Singular integral operator
- Space of homogeneous type
- Sparse domination