We show that a function f : R → R of bounded variation satisfies VarMf ≤ C Var f, where Mf is the centered Hardy–Littlewood maximal function of f . Consequently, the operator f 7→ (Mf) is bounded from W (R) to L(R). This answers a question of Hajłasz and Onninen in the one-dimensional case.

Let L be the space of complex-valued functions f on the set of vertices T of an infinite tree rooted at o such that the difference of the values of f at neighboring vertices remains bounded throughout the tree, and let Lw be the set of functions f ∈ L such that |f v − f v− | O |v|−1 , where |v| is the distance between o and v and v− is the neighbor of v closest to o. In this paper, we character...

A function f of a graph is called a complete graph invariant if G ∼= H is equivalent to f(G) = f(H). If, in addition, f(G) is a graph isomorphic to G, then f is called a canonical form for graphs. Gurevich [6] proves that graphs have a polynomial-time computable canonical form exactly when they have a polynomial-time computable complete invariant. We extend this equivalence to the polylogarithm...

Let F be a field and m,n be integers m,n > 3. Let SMn(F) and STn(F) denote the linear space of n × n per-symmetric matrices over F and the linear space of n × n per-symmetric triangular matrices over F, respectively. In this note, the structure of spaces of bounded rank-two matrices of STn(F) is determined. Using this structural result, a classification of bounded rank-two linear preservers ψ :...

Let F : M → N be a harmonic map between complete Riemannian manifolds. Assume that N is simply connected with sectional curvature bounded between two negative constants. If F is a quasiconformal harmonic diffeomorphism, then M supports an infinite dimensional space of bounded harmonic functions. On the other hand, if M supports no non-constant bounded harmonic functions, then any harmonic map o...

for some real number X, the symbol V denoting the lattice least upper bound. Any ring R is regular [10] if for each xER there is an xaER such that xx°x = x. It is evident that every regular F-ring R contains a maximal bounded sub-F-ring R, the F-ring of all xER satisfying equation (1.1). The relationship between a regular F-ring and its maximal bounded sub-F-ring is analogous to that between th...

Let D be the open unit disk in the complex plane C. Denote by H(D) the class of all functions analytic on D. An analytic self-map φ : D → D induces the composition operator Cφ on H(D), defined by Cφ ( f ) = f (φ(z)) for f analytic on D. It is a well-known consequence of Littlewood’s subordination principle that the composition operator Cφ is bounded on the classical Hardy and Bergman spaces (se...

in the region t>0. Here x =(Xl, x2 . . . . , xn), u =u(t, x), the f j , j = 1, 2, . . . , n, are continuously differentiable functions of a single real variable, and u0 (x) is in the class F; i.e., Uo(X) is a bounded function having locally bounded variation in the sense of TOt~LLI-CESARI. Moreover for each fixed t, u(t, x) is in the same class F. A function f is said to have bounded variation ...

The terminology and notation used here are introduced in the following articles: [18], [8], [20], [5], [7], [6], [3], [1], [17], [13], [19], [14], [2], [4], [15], [10], [11], [9], and [12]. One can prove the following propositions: (1) Let X, Y , Z be complex linear spaces, f be a linear operator from X into Y , and g be a linear operator from Y into Z. Then g · f is a linear operator from X in...

We join Cardelli's and Mitchell's approaches to polymorphic subtyping by introducing a new uniied calculus G that subsumes both Cardelli-inspired systems with rigid subtyping on the one hand, and the more liberal Mitchell's containment calculus F Mit88] on the other. Up until now both approaches were considered incompatible, since F-provable containments between functional and universal types a...