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Integral operators with variable kernels on weak Hardy spaces. (English) Zbl 1099.47040

The authors study the mapping properties for the singular integral operators of Calderón–Zygmund type \[ T_{\Omega,\alpha}f(x)=\int_{R^n}\frac{\Omega(x,x-y)}{| x-y| ^{n-\alpha}} f(y) dy,\quad 0\leq\alpha<n, \] where the characteristic \(\Omega(x,y)\) is bounded in the first variable and homogeneous of zero degree and \(q\)-integrable over the unit sphere in the second variable (with the standard cancellation condition). In particular, the authors study the mapping properties of the operators \(T_{\Omega,\alpha}\) on the so-called weak Hardy spaces, introduced in [Ch. Fefferman and F. Soria, Stud.Math.85, 1–16 (1986; Zbl 0626.42013)]. They provide corresponding sufficient conditions for the mentioned mapping in terms of the order \(q\) modulus of continuity of the characteristic function \(\Omega(x,y)\), namely: \[ \omega_q(\delta)= \sup_{\| \rho\| \leq \delta}\left(\int_{S^{n-1}}\sup_{x\in R^n}| \Omega(x,\rho z/| z| )-\Omega(x, z/| z| )| ^q d\sigma (z/| z| )\right)^{1/q}. \]

MSC:

47G10 Integral operators
42B35 Function spaces arising in harmonic analysis
42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)

Citations:

Zbl 0626.42013
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References:

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