This paper considers analogs of results on integral operators studied by Hörmander. Using the sharp function introduced by Fefferman and Stein, we prove weighted norm inequalities on kernel operators which map an Lp space into an Lq space, with q not equal to p. The techniques recover known results about fractional integral operators and apply to multiplier operators which satisfy a generalization of the Hörmander multiplier condition.

Let D ⊂ Rn be a bounded domain and L: dom L ⊂ L2 (D) → L2 (D) be a self-adjoint operator of finite dimensional kernel. Let f: D × R → R be a function satisfying the Carathéodory condition. Assume that there are constants λ > 0 and δ ∈ [0, 1] such that and that .

Then with the aid of a generalized Krasnosel'skii's theorem it has been proved that under conditions exactly analogous to those of Landesman and Lazer there exists u ∈ L2(D) such that L(u)(x) = f(x, u(x)) for ∀x ∈ D. This result is then used to prove the existence of weak solutions of nonlinesr elliptic boundary value problems.

Other abstract results applicable to ordinary and partial differential equations have also been proved.

In this paper, we are concerned with certain generalizations of subnormal and ascendent (transfinitely subnormal) subgroups of a group. A subgroup A of a group G is called f-ascendent in G if there is a well ordered ascending complete series of subgroups of G, where for all α < λ, either Gα ⊲ Gα+1 or [Gα+1: Gα] < ∞. If such a series has finite length, A is called F-subnormal in G.

Let H denote a set with three elements, and T3 the full transformation semigroup on X, i.e. T3 consists of the twenty-seven self maps of X under functional composition. A transformation semigroup (briefly a τ-semigroup) on three letters is an ordered pair (X, S), where S is any subsemigroup of T3.

This paper introduces two new separation axioms, point paracompactness and point countable paracompactness, both somewhat weaker than regularity, and shows that they can replace regularity in several standard theorems about paracompact or absolutely H-closed or Lindelöf spaces. Thus we obtain sharpened versions of these theorems. We also show that under certain hypotheses the new properties are equivalent to regularity.

Let R be a not necessarily commutative local ring, M a free R-module, and π ∈ GL(M) such that B(π) = im(π –1)is a subspace of M. Then π = σ1…σtρ, where σi are simple mappings of given types, ρ is a simple mapping, B(sgr;i) and B(ρ) are subspaces and t ≤ dim B(π).

We give an alternative short proof of a recent theorem of J. A. Hillman and P.A. Linnell that an elementary amenable group with finite Hirsch number has, modulo its locally finite radical, a soluble normal subgroup with index and derived length bounded only in terms of the Hirsch number of the group.

We give continued fraction algorithms for each conjugacy class of triangle Fuchsian group of signature $(3, n, \infty )$, with $n\geq 4$. In particular, we give an explicit form of the group that is a subgroup of the Hilbert modular group of its trace field and provide an interval map that is piecewise linear fractional, given in terms of group elements. Using natural extensions, we find an ergodic invariant measure for the interval map. We also study Diophantine properties of approximation in terms of the continued fractions and show that these continued fractions are appropriate to obtain transcendence results.

If two functions of a real variable are integrable over two intervals, say of t, τ, respectively, then the product of the two functions should be integrable over the rectangular product of the two intervals of t and τ. For the Lebesgue integral, definable using non-negative functions alone, the proof is easy. For non-absolute integrals such as the Perron, Çesàro-Perron, and Marcinkiewicz-Zygmund integrals we have difficulties since the functions cannot be assumed non-negative. But the present paper gives a proof.

The usual technique for dealing with an abelian W*-algebra is to consider it, via the Gelfand theory, as the algebra of all continuous complex-valued functions on an extremally disconnected compact Hausdorff space with a separating family of normal linear functionals. An alternative approach, outlined in [2] and [10], is to develop the theory within the framework of Riesz spaces (linear vector lattices) where the order properties of the self-adjoint operators play an important and natural role. It has been known for a long time that the self-adjoint part of an abelian W*-algebra is a Dedekind complete Riesz space under the natural ordering of self-adjoint operators, but it is only relatively recently that a proof of this fact has been given that is independent of the Gelfand theory, and the interested reader may consult [2] or [10] for the details. This approach is essentially foreshadowed in [6] and provides a very satisfying introduction to the theory of commutative rings of operators. From this point of view, the spectral theorem for self-adjoint operators falls naturally into place as an easy consequence of the spectral theorem of H. Freudenthal. In this paper, the line of approach via Riesz spaces is developed further and several well known results are shown to follow as elementary consequences of the order structure of the algebra.

The group near-ring constructed from a right near-ring R and a group G is studied in the special case where the near-ring is distributively generated. In particular, results concerning homomorphisms of near-rings or of groups and the augmentation ideal are obtained which resemble closely those obtained for group rings.

Let E be a vector lattice in the sense of [2]. Two elements a, b in E are said to be disjoint (written a⊥b) if ′a′^′b′ = 0. For any subset A of E, A⊥ is the set If A is a one point set {a}, then we shall use a⊥ for {a}⊥. A⊥⊥ is denned by A⊥⊥ = {A⊥}⊥. A subset B of E is said to be a polar if it is of the form A⊥ for some subset A of E. It is well known [1] that the set of all polars of E ordered by inclusion forms a Boolean algebra. We shall denote this algebra by B, and M will mean the set of all maximal ideals of B. The following properties of polars will be used in the paper.

A characterization is provided here for any normal partition of the set of idempotents of a regular semigroup S. As a by-product of the method used, a new characterization of the greatest congruence on S corresponding to a given normal partition of its idempotents is obtained.

Subject classification (Amer. Math. Soc. (MOS) 1970): 20 M 10.

We consider the lattice of pseudovarieties contained in a given pseudovariety P. It is shown that if the lattice L of subpseudovarieties of P has finite height, then L is isomorphic to the lattice of subvarieties of a locally finite variety. Thus not every finite lattice is isomorphic to a lattice of subpseudovarieties. Moreover, the lattice of subpseudovarieties of P satisfies every positive universal sentence holding in all lattice of subvarieties of varieties V(A) ganarated by algebras A ε P.

It is shown that all the generalized Whitehead products vanish in X and all the components of XΣA have the same homotopy type when X is a T-space. It is also shown that any T-space is a G-space. The dual spaces of T-spaces are introduced and studied.

The existence of 1-factorizations of the complete multigraph λKn which cannot be decomposed into 1-factorizations with smaller λ is studied.

Solutions of the iterated equation of generalized axially symmetric potential theory [1]

where the operator Lk is defined by

will be denoted by except that when n = 1, fk will be written instead of . It is easily shown [2, 3] that

by which is meant that any function is a solution of (1).

We prove that if two normed-algebra-valued cosine families indexed by a single Abelian group, of which one is bounded and comprised solely of scalar elements of the underlying algebra, differ in norm by less than $1$ uniformly in the parametrising index, then these families coincide.

Throughout this paper only abelian l-groups will be considered and G will denote an abelian l-group. G is large in the l-group H or H is an essential extension of G if G is an l-subgroup of H and for each l-ideal L ≠ 0 of H we have L ∩ G ≠ 0. A ν-hull of G is a minimal vector lattice that contains G and is an essential extension of G. Each G admits a ν-hull (Conrad (1970)).We shall be interested in the following properties of G.

A (v, k, λ) configurations is conjectured to exist for every v, k and λ satisfying λ(v − 1) = k(k − 1) and k − λ is a square if v is even, x2 = (k − λ)y2 + (−1)(v−1)/2 λz2 has a solution in integers x, y and z not all zero for v odd.