Wavelets and the well-posedness of incompressible magneto-hydrodynamic equations in Besov type<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" display="inline" overflow="scroll"><mml:mi>Q</mml:mi></mml:math>-space

Let (, d, ) be a metric measure space of homogeneous type in the sense of Coifman and Weiss. In this article, the authors prove that bilinear operators, which are finite combinations of compositions of commutators and Calderón-Zygmund operators, are bounded from H 1 ()×BMO() to L 1 (). The authors also prove that the commutator, generated by any b ∈ BMO() and Calderón-Zygmund operator, is bounded from the Hardy-type spacethat ensures the boundedness of the commutators from  to L 1 (). The novelties appearing in these approaches exist in applications of the multiresolution analysis of the wavelets on metric measure spaces of homogeneous type, the bilinear decomposition of the product space H 1 () × BMO(), the (sub)bilinear decomposition of commutators, the proof of off-diagonal estimates of the action of Calderón-Zygmund operators on the wavelet functions, and the boundedness of the almost diagonal matrix on the spaces H 1 () and BMO(). Notably, throughout this article, is not assumed to satisfy the reverse doubling condition. KEYWORDSbilinear decomposition, BMO space, Calderón-Zygmund operator, commutator, Hardy space, space of homogeneous type, wavelet for the theory of singular integrals and function spaces. A metric space (, d) is a set  equipped with a metric d such that, for any x, , z ∈ , (i) d(x, y) = d(y, x);

Section: Introduction and Main Resultsmentioning

confidence: 99%

Endpoint estimates of linear commutators on Hardy spaces over spaces of homogeneous type

Liu

Chang

et al. 2018

“…Conversely, we assume that v satisfies (9). For any fixed ( , s) ∈ R n+1 + , let Q be the cube parallel to the coordinate axes, centered at y, and with the edge length 2s.…”

Section: Carleson Measures Characterizationmentioning

confidence: 99%

“…In recent years,  , (R n ) has been studied extensively; see Cui-Yang, 1 Li-Xiao-Yang, 2 Liang-Sawano-Ullrich-Yang-Yuan, 3 Strichartz, 4 Wang-Xiao, 5 Yang-Yuan, 6 Yuan-Sickel-Yang, 7 and the references therein. For the applications of  , (R n ) on PDE, we refer the reader to Li-Xiao-Yang, 2 Li-Yang, 8,9 and Lin-Yang. 10 It is well known that the Poisson integral describes the relations between the harmonic function spaces on R n+1 + and their boundary values.…”

Section: Introductionmentioning

confidence: 99%

Poisson semigroup characterization and the trace theorem of a class of fractional mean oscillation spaces

Peng

2018

Self Cite

In this paper, we use regular wavelets to study the Poisson extension of the fractional mean oscillation spaces  , (R n ). Via a distributional trace operator , we establish a relation between  , (R n ) and a class of harmonic function spaces  , (R n+1 + ). KEYWORDSdistributional trace, fractional bounded mean oscillation spaces, harmonic function, waveletwhere the supremum is taken over all cubes Q and S ,p,f denotes the set of all polynomials satisfying certain conditions; see Definition 2.4 for the details. The spaces  , (R n ) can be seen as the generalizations of the following function spaces: the bounded mean oscillation space BMO(R n ), the Bloch space B 0,∞ ∞ (R n ), the Hölder spaces  (R n ), 0 < < 1, and the Besov-Q spacesḂ ,0 , (R n ). We give the following table to clarify the relation between these spaces and  , (R n ):

“…The Fourier transform of a function on Heisenberg group is operator‐valued (see and ), which is defined by

\hat{f} (λ) = \int_{H^{d}} f (z, t) π_{λ} (z, t) dzdt,

where

π_{λ} (z, t) = e^{- iλt} \exp (- z W_{λ} + \overset{z}{false} W_{λ}^{+})

is the irreducible unitary representation of

{double-struckH}^{d}

. We have the Plancherel formula

∥ f ∥_{L^{2} (H^{d})} = {\{\frac{2^{d - 1}}{π^{d + 1}} \int_{- \infty}^{+ \infty} ∥ \hat{f} (λ) ∥_{H - S}^{2} | λ |^{d} dλ\}}^{\frac{1}{2}}, f \in L^{1} (H^{d}) ⋂ L^{2} (H^{d}),

where ∥·∥ H − S denotes the Hilbert–Schmidt norm of operators.…”

Section: Preliminaries On Heisenberg Group Double-struckhdmentioning

confidence: 99%

“…In recent years, such wavelet characterizations of Besov type spaces have been applied to the nonlinear problems in fluid equations. See Lin and Yang , Li and Yang and Li et al . .…”

Section: Introductionmentioning

confidence: 99%

Regular orthogonal basis on Heisenberg group and application to function spaces

Yang

2014

Self Cite

In this paper, we construct some compactly supported orthogonal regular wavelet basis on Heisenberg group H d . Because of the regularity of wavelets, we could use such wavelets to characterize function spaces on H d , such as bounded mean oscillation space (BMO) space, Hardy space, Besov spaces and Besov-Morrey spaces.where S.z, z 0 / D 2Q szN z 0 is the symplectic inner product (or symplectic form). Alternatively, by the real coordinates, the aforementioned multiplication can be represented as .x, y, t/.x 0 , y 0 , t 0 / D .x C x 0 , y C y 0 , t C t 0 C 2.y 0 x x 0 y//, (2.2)