Maximal estimates for the (C, alpha) means of d-dimensional Walsh-Fourier series

The d-dimensional dyadic martingale Hardy spaces H(p) are introduced and it is proved that the maximal operator of the (C, alpha) (alpha = (alpha(1),..., alpha(d))) means of a Walsh-Fourier series is bounded from H(p) to L(p) (1/(alpha(k) + 1) < p < infinity) and is of weak type(L(1), L(1)), provided that the supremum in the maximal operator is taken over a positive cone. As a consequence we obtain that the (C, alpha) means of a function f is an element of L(1) converge a.e. to the function in question. Moreover, we prove that the (C; alpha) means are uniformly bounded on H(p) whenever 1/(alpha(k) + 1)< p < infinity. Thus, in case f is an element of H(p), the (C; alpha) means converge to f in H(p) norm. The same results are proved for the conjugate (C; alpha) means, too

A generalization for Fourier transforms of a theorem due to Marcinkiewicz

It is shown that the maximal operator of the Marcinkiewicz means of a tempered distribution is bounded from H(p)(R(2)) to L(p)(R(2)) for all p(0) infinity and, consequently, is of weak type (1, 1), where p(0) L(1)(R(2)) converge a.e. to the function in question. Moreover, we prove that the Marcinkiewicz means are uniformly bounded on the spaces H(p)(R(2)) and so they converge in the norm (p(0) < p < infinity). Similar results for the Riesz transforms are also given. (C) 2000 Academic Press

Static quark anti-quark free energy and the running coupling at finite temperature

We analyze the free energy of a static quark anti-quark pair in quenched QCD at short and large distances. From this we deduce running couplings, g^2(r,T), and determine the length scale that separates at high temperature the short distance perturbative regime from the large distance non-perturbative regime in the QCD plasma phase. Ambiguities in the definition of a coupling beyond the perturbative regime are discussed in their relation to phenomenological considerations on heavy quark bound states in the quark gluon plasma. Our analysis suggests that it is more appropriate to characterize the non-perturbative properties of the QCD plasma phase close to T_c in terms remnants of the confinement part of the QCD force rather than a strong Coulombic force.Comment: 8 pages, 9 EPS-files, revtex

Thermodynamics of Four-Flavour QCD with Improved Staggered Fermions

We have calculated the pressure and energy density in four-flavour QCD using improved fermion and gauge actions. We observe a strong reduction of finite cut-off effects in the high temperature regime, similar to what has been noted before for the SU(3) gauge theory. Calculations have been performed on $16^3\times 4$ and 16^4 lattices for two values of the quark mass, $ma = 0.05$ and 0.1. A calculation of the string tension at zero temperature yields a critical temperature $T_c/\sqrt{\sigma} = 0.407 \pm 0.010$ for the smaller quark mass value.Comment: 12 pages, LaTeX2e File, 11 encapsulated postscript file

Drastic Reduction of Cutoff Effects in 2-d Lattice O(N) Models

We investigate the cutoff effects in 2-d lattice O(N) models for a variety of lattice actions, and we identify a class of very simple actions for which the lattice artifacts are extremely small. One action agrees with the standard action, except that it constrains neighboring spins to a maximal relative angle delta. We fix delta by demanding that a particular value of the step scaling function agrees with its continuum result already on a rather coarse lattice. Remarkably, the cutoff effects of the entire step scaling function are then reduced to the per mille level. This also applies to the theta-vacuum effects of the step scaling function in the 2-d O(3) model. The cutoff effects of other physical observables including the renormalized coupling and the mass in the isotensor channel are also reduced drastically. Another choice, the mixed action, which combines the standard quadratic with an appropriately tuned large quartic term, also has extremely small cutoff effects. The size of cutoff effects is also investigated analytically in 1-d and at N = infinity in 2-d.Comment: 39 pages, 18 figure

Improved Actions for QCD Thermodynamics on the Lattice

Finite cut-off effects strongly influence the thermodynamics of lattice regularized QCD at high temperature in the standard Wilson formulation. We analyze the reduction of finite cut-off effects in formulations of the thermodynamics of $SU(N)$ gauge theories with three different $O(a^2)$ and $O(a^4)$ improved actions. We calculate the energy density and pressure on finite lattices in leading order weak coupling perturbation theory ($T\rightarrow \infty$) and perform Monte Carlo simulations with improved $SU(3)$ actions at non-zero $g^2$. Already on lattices with temporal extent $N_\tau=4$ we find a strong reduction of finite cut-off effects in the high temperature limit, which persists also down to temperatures a few times the deconfinement transition temperature.Comment: 20 pages, 3 Postscript figure

Logarithmic corrections to O(a(2)) lattice artifacts

We compute logarithmic corrections to the O(a(2)) lattice artifacts for a class of lattice actions for the non-linear O(n) sigma-model in two dimensions. The generic leading artifacts are of the form a(2)vertical bar In(a(2))vertical bar(n/(n-2)). We also compute the next-to-leading corrections and show that for the case n = 3 the resulting expressions describe well the lattice artifacts in the step scaling function, which are in a large range of the cutoff apparently of the form O(a). An analogous computation should, if technically possible, accompany ally precision measurements in lattice QCD. (C) 2009 Elsevier B.V. All rights reserved