We introduce multilinear analogues of dyadic paraproduct operators and Haar Multipliers, and study boundedness properties of these operators and their commutators. We also characterize dyadic BM O functions via boundedness of certain paraproducts and also via boundedness of the commutators of multilinear Haar Multipliers and paraproduct operators.
Abstract. We introduce multilinear analogues of dyadic paraproduct operators and Haar Multipliers, and study boundedness properties of these operators and their commutators. We also characterize dyadic BM O functions via boundedness of certain paraproducts and also via boundedness of the commutators of multilinear Haar Multipliers and paraproduct operators. Introduction and statement of main resultsDyadic operators have attracted a lot of attention in the recent years. The proof of so-called A 2 theorem (see [7]) consisted in representing a general Calderón-Zygmund operator as an average of dyadic shifts, and then verifying some testing conditions for those simpler dyadic operators. It seems reasonable to believe that, taking a similar approach, general multilinear Calderón-Zygmund operators can be studied by studying multilinear dyadic operators. Regardless of this possibility, multilinear dyadic operators in their own right are an important class of objects in Harmonic Analysis. Statements regarding those operators can be translated into the non-dyadic world, and are sometimes simpler to prove.In this paper we introduce multilinear analogues of dyadic operators such as paraproducts and Haar multipliers, and study their boundedness properties. Corresponding theory of linear dyadic operators, which we will be using very often, can be found in [11]. In [1], the authors have studied boundedness properties of bilinear paraproducts defined in terms of so-called "smooth molecules". The paraproduct operators we study are more general multilinear operators, but defined in terms of indicators and Haar functions of dyadic intervals. In [3] Coifman, Rochberg and Weiss proved that the commutator of a BMO function with a singular integral operator is bounded in L p , 1 < p < ∞. The necessity of BMO condition for the boundedness of the commutator was also established for certain singular integral operators, such as the Hilbert transform. S. Janson [8] later studied its analogue for linear 2000 Mathematics Subject Classification. Primary .
Let \(X\) be a real reflexive Banach space and \(X^*\) be its dual space. Let \(G_1\) and \(G_2\) be open subsets of \(X\) such that \(\overline G_2\subset G_1\), \(0\in G_2\), and \(G_1\) is bounded. Let \(L: X\supset D(L)\to X^*\) be a densely defined linear maximal monotone operator, \(A:X\supset D(A)\to 2^{X^*}\) be a maximal monotone and positively homogeneous operator of degree \(\gamma>0\), \(C:X\supset D(C)\to X^*\) be a bounded demicontinuous operator of type \((S_+)\) with respect to \(D(L)\), and \(T:\overline G_1\to 2^{X^*}\) be a compact and upper-semicontinuous operator whose alues are closed and convex sets in \(X^*\). We first take \(L=0\) and establish the existence of nonzero solutions of \(Ax+ Cx+ Tx\ni 0\) in the set \(G_1\setminus G_2\). Secondly, we assume that \(A\) is bounded and establish the existence of nonzero solutions of \(Lx+Ax+Cx\ni 0\) in \(G_1\setminus G_2\). We remove the restrictions \(\gamma\in (0, 1]\) for \(Ax+ Cx+ Tx\ni 0\) and \(\gamma= 1\) for \(Lx+Ax+Cx\ni 0\) from such existing results in the literature. We also present applications to elliptic and parabolic partial differential equations in general divergence form satisfying Dirichlet boundary conditions.
Let D be an open subset of R N and f : D → R N a continuous function. The classical topological degree for f demands that D be bounded. The boundedness of domains is also assumed for the topological degrees for compact displacements of the identity and for operators of monotone type in Banach spaces. In this work, we follow the methodology introduced by Nagumo for constructing topological degrees for functions on unbounded domains in finite dimensions and define the degrees for Leray-Schauder operators and (S + )-operators on unbounded domains in infinite dimensions.
Let X be a real reflexive Banach space with X * its dual space. Let L : X ⊃ D(L) → X * be a densely defined linear maximal monotone operator, A : X ⊃ D(A) → 2 X * be a maximal monotone and positively homogeneous operator of degree γ > 0, C : X ⊃ D(C) → X * be a bounded demicontinuous operator of type (S + ) w.r.t. D(L), and Q : G 1 → 2 X * be a compact and upper-semicontinuous operator whose values are closed and convex sets in X * . In the case L = 0, we establish the existence of nonzero solutions ofand G 1 is bounded. Otherwise, we assume that A is bounded and establish the existence of nonzero solutions of Lx + Ax + Cx ∋ 0 in the set G 1 \ G 2 as above. We completely remove the restrictions γ ∈ (0, 1] for Ax + Cx + Qx ∋ 0 and γ = 1 for Lx + Ax + Cx ∋ 0 from these previous results established by the first author. Applications to elliptic and parabolic partial differential inclusions in general divergence form that include the p-Laplacian with 1 < p < ∞ and satisfy Dirichlet boundary conditions are also presented.
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