A boundedness result for Marcinkiewicz integral operator

We extend a boundedness result for Marcinkiewicz integral operator. We find a new space of radial functions for which this class of singular integral operators remains {L}^{p} -bounded when its kernel satisfies only the sole integrability condition.

Higher-Order Commutators of Parametric Marcinkiewicz Integrals on Herz Spaces with Variable Exponent

LetΩ∈Ls(Sn-1)fors⩾1be a homogeneous function of degree zero andbbe BMO functions. In this paper, we obtain some boundedness of the parametric Marcinkiewicz integral operatorμΩρand its higher-order commutator[bm,μΩρ]on Herz spaces with variable exponent.

GENERALIZED MORREY SPACES OVER NONHOMOGENEOUS METRIC MEASURE SPACES

Let $({\mathcal{X}},d,\unicode[STIX]{x1D707})$ be a nonhomogeneous metric measure space satisfying the so-called upper doubling and the geometric doubling conditions. In this paper, the authors give the natural definition of the generalized Morrey spaces on $({\mathcal{X}},d,\unicode[STIX]{x1D707})$, and then investigate some properties of the maximal operator, the fractional integral operator and its commutator, and the Marcinkiewicz integral operator.

The Boundedness of Marcinkiewicz Integral Associated with Schrödinger Operator and Its Commutator

The authors prove that Marcinkiewicz integral operator is not only are bounded onLp, for1<P<∞, but also a bounded map fromL1(Rn)to weakL1(Rn). Meanwhile, theBMOL-boundedness and(HL1,L1)-boundedness are also obtained. Finally, theLp-boundedness and(L∞,BMOL)-boundedness for the commutator of Marcinkiewicz integral of schrödinger type are established.

ESTIMATES FOR MARCINKIEWICZ INTEGRALS IN BMO AND CAMPANATO SPACES

In this paper, the authors consider the behavior on BMO($\mathbb R^n$) and Campanato spaces for the higher-dimensional Marcinkiewicz integral operator which is defined by where Ω is homogeneous of degree zero, has mean value zero and is integrable on the unit sphere. Under certain weak regularity condition on Ω, the authors prove that if f belongs to BMO($\mathbb R^n$) or to a certain Campanato space, then [μΩ(f)]2 is either infinite everywhere or finite almost everywhere, and in the latter case, some kind of boundedness is also obtained. The corresponding Lusin area integral is also considered.

Marcinkiewicz integral operators on product domains

With the cancellation property of the bounded kernel, we prove that the generalized Marcinkiewicz integral operator is bounded on L2 (ℝn×ℝm) for all dimensions n, m.