Abstract. In this paper, we provide a version of the Mihlin-Hörmander multiplier theorem for multilinear operators in the case where the target space is L p for p ≤ 1. This extends a recent result of Tomita [15] who proved an analogous result for p > 1.

We establish the Triebel-Lizorkin boundedness for a class of singular integral operators associated to surfaces of revolution, {( , ( )) : ∈ ℝ }, with rough kernels Ω, provided that the corresponding maximal function along the plane curve {( , ( )) : ∈ ℝ} is bounded on (ℝ 2 ). We treat kernel functions belonging to a generalized function space relating to Grafakos and Stefanov's F ( −1 ) spaces.

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LetLbe the infinitesimal generator of an analytic semigroup onL2(Rn)with Gaussian kernel bounds, and letL-α/2be the fractional integrals ofLfor0<α<n. For any locally integrable functionb, the commutators associated withL-α/2are defined by[b,L-α/2](f)(x)=b(x)L-α/2(f)(x)-L-α/2(bf)(x). Whenb∈BMO(ω)(weightedBMOspace) orb∈BMO, the authors obtain the necessary and sufficient conditions for the boundedness of[b,L-α/2]on weighted Morrey spaces, respectively.

Let 0 < α < n and M α be the fractional maximal function. The nonlinear commutator of M α and a locally integrable function b is given by [b, M α ](f) = bM α (f)-M α (bf). In this paper, we mainly give some necessary and sufficient conditions for the boundedness of [b, M α ] on variable Lebesgue spaces when b belongs to Lipschitz or BMO(R n) spaces, by which some new characterizations for certain subclasses of Lipschitz and BMO(R n) spaces are obtained.

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Let L be a linear operator in L 2 (R n ) which generates a semigroup e −tL whose kernels p t (x, y) satisfy the Gaussian upper bound. In this paper, we investigate several kinds of weighted norm inequalities for the conical square function S α,L associated with an abstract operator L. We first establish two-weight inequalities including bump estimates, and Fefferman-Stein inequalities with arbitrary weights. We also present the local decay estimates using the extrapolation techniques, and the mixed weak type estimates corresponding Sawyer's conjecture by means of a Coifman-Fefferman inequality. Beyond that, we consider other weak type estimates including the restricted weak-type (p, p) for S α,L and the endpoint estimate for commutators of S α,L . Finally, all the conclusions aforementioned can be applied to a number of square functions associated to L.

LetTbe a multilinear square function with a kernel satisfying Dini(1) condition and letT⁎be the corresponding multilinear maximal square function. In this paper, first, we showed thatTis bounded fromL1×⋯×L1toL1/m,∞.Secondly, we obtained that if eachpi>1, thenTandT⁎are bounded fromLp1(ω1)×⋯×Lpm(ωm)toLp(νω→)and if there ispi=1, thenTandT⁎are bounded fromLp1(ω1)×⋯×Lpm(ωm)toLp,∞(νω→), whereνω→=∏i=1mωip/pi.Furthermore, we established the weighted strong and weak type boundedness forTandT⁎on weighted Morrey type spaces, respectively.