Embedding lp, {infty} into Lp, {infty}[0, 1] Complementably

We show that l^p, embeds into L^p, [0, 1] as a complemented sublattice.     

Heat Kernel Estimates and LP Spectral Independence of Elliptic Operators

Let be an open subset of R^d, and let T_p for p[1, ) be consistentC₀-semigroups given by kernels that satisfy an upper heat kernelestimate. Denoting their generators by A_p, we show that thespectrum (A_p) is independent of p[1, ). We also treat the caseof weighted L^p-spaces for weights that satisfy a subexponentialgrowth condition. An example shows that independence of thespectrum may fail for an exponential weight. 1991 MathematicsSubject Classification 47D06, 47A10, 35P05.     

The Direct Limits Of The Banach-mazur Compacta

Let 1 p . For each n-dimensional Banach space E = (E, || ·||), we define a norm || · ||_p on E x R as follows: [formula] It is shown that the correspondence (E, || · ||) (Ex R, || · ||_p) defines a topological embedding of oneBanach–Mazur compactum, BM(n), into another, BM(n 1),and hence we obtain a tower of Banach–Mazur compacta:BM(1) BM(2) BM(3) ···. Let BM_p be thedirect limit of this tower. We prove that BM_p is homeomorphicto Q = dir lim Qⁿ, where Q = [0, 1] is the Hilbert cube. 1991Mathematics Subject Classification 46B04, 46B20, 52A21, 57N20,54H15.     

Lower Bounds for the Lp-Norm in Terms of the Mellin Transform

Given a measurable function f on (0, ) with Mellin transformF(s), let |f|_p denote the L^p-norm of f with respect to the measuredx/x. We prove that under certain assumptions, for instanceif f is real and non-negative and F() converges for in an openinterval and F() 0, then wherec_p (2e)^–1. We derive similar inequalities for complex-valuedf, for the L^p-norm of the derivative of f, and for the supremumof real-valued f and of its derivative. The lower bounds areeminently applicable when f is a convolution product.     

A Comparison Estimate for the Heat Equation with an Application to the Heat Content of the S-Adic Von Koch Snowflake

Let D be an open set in Euclidean space R^m with boundary D,and let :D[0, ) be a bounded, measurable function. Let u:DDx[0,)[0, ) be the unique weak solution of the heat equation [formula] with initial condition [formula] and with inhomogeneous Dirichlet boundary condition [formula] Then u(x; t) represents the temperature at a point xD at timet if D has initial temperature 0, while the temperature at apoint xD is kept fixed at (x) for all t>0. We define thetotal heat content (or energy) in D at time t by [formula] In this paper we wish to examine the effect of imposing additionalcooling on some subset C on both u and E_D. 1991 MathematicsSubject Classification 35K05, 60J65, 28A80.     

A problem of Hirst on continued fractions with sequences of partial quotients

Let B denote an infinite sequence of positive integers b₁ <b₂ < ..., and let denote the exponent of convergence ofthe series _{n = 1} 1/b_n; that is, = inf {s 0 : _{n = 1} 1/b_n^s <}. Define E(B) = {x [0, 1]: a_n(x) B (n 1) and a_n(x) asn }. K. E. Hirst [Proc. Amer. Math. Soc. 38 (1973) 221–227]proved the inequality dim_H E(B) /2 and conjectured (see ibid.,p. 225 and [T. W. Cusick, Quart. J. Math. Oxford (2) 41 (1990)p. 278]) that equality holds. In this paper, we give a positiveanswer to this conjecture.     

A Relation between Hausdorff Dimension and a Condition on Time Sets for the Image by the Brownian Sheet to Possess Interior Points

We prove that if W_{N, d} is a Brownian sheet mapping to R^d and E is a set in (0, )^N of Hausdorff dimensiongreater than , then for almost every rotation about a point x and translation _x such that _x(E) (0, )^N, the set _x(E) is such that almost surely W(E) containsinterior points. The techniques are adapted from Kahane andRosen and generalize to higher dimensional time and range.     

Sharp Distortion Estimate for Locally Schlicht Bloch Functions

Let B be the space of locally schlicht Bloch functions f whichare analytic in the unit disc with f(0) = f'(0) – 1 =0 satisfying 0 < |f'(z)|(1 – |z|²) 1. For each fixedz₀ we shall determine the shape of the set {logf'(z₀): fB},that is, we shall give the sharp distortion estimate for locallyschilcht Bloch functions.     

An Example on Sobolev Space Approximation

For each d2 we construct a connected open set R^d such that = int (clos()), and for each k 1 and each p [1, ), the subsetW^k, () fails to be dense in the Sobolev space W^{k, p}(), in thenorm of W^{k, p}(). 1991 Mathematics Subject Classification 46E35,46F05.     

Cluster Sets of Harmonic Functions at the Boundary of a Half-Space

The purpose of this paper is to answer some questions posedby Doob [2] in 1965 concerning the boundary cluster sets ofharmonic and superharmonic functions on the half-space D givenby D = R^n–1 x (0, + ), where n 2. Let f: D [–,+] and let Z D. Following Doob, we write B_Z (respectively C_Z)for the non-tangential (respectively minimal fine) cluster setof f at Z. Thus l B_Z if and only if there is a sequence (X_m)of points in D which approaches Z non-tangentially and satisfiesf(X_m) l. Also, l C_Z if and only if there is a subset E ofD which is not minimally thin at Z with respect to D, and whichsatisfies f(X) l as X Z along E. (We refer to the book byDoob [3, 1.XII] for an account of the minimal fine topology.In particular, the latter equivalence may be found in [3, 1.XII.16].)If f is superharmonic on D, then (see [2, 6]) both sets B_Z andC_Z are subintervals of [–, +]. Let denote (n –1)-dimensional measure on D. The following results are due toDoob [2, Theorem 6.1 and p. 123]. 1991 Mathematics Subject Classification31B25.     

On the Best Constant in Weighted Inequalities for Riemann-Liouville Integrals

For 1 k < and 1 p q , the problem of finding the bestconstant C_pq in the weighted inequality involving the Riemann-Liouville integrals of theform is considered.     

On Finite and InfiniteCk-A-Determinacy

Let f: (Rⁿ,0) (R^p,0) be a C map-germ. We define f to be finitely,or -, A-determined, if there exists an integer m such that allgerms g with j^mg(0) = j^mf(0), or if all germs g with the sameinfinite Taylor series as f, respectively, are A-equivalentto f. For any integer k, 0 k < , we can consider A' sC^kcounterpart (consisting of C^k diffeomorphisms) A^(k), and wecan define the notion of finite, or -,A^(k)-determinacy in asimilar manner. Consider the following conditions for a C germf: (a_k) f is -A^(k)-determined, (b_k) f is finitely A^(k)-determined,(t) , (g) there exists a representative f : U R^p defined on some neighbourhood U of 0 in Rⁿ such thatthe multigerm of f is stable at every finite set , and (g') every f' with j f'(0)=j f(0) satisfiescondition (g). We also define a technical condition which willimply condition (g) above. This condition is a collection ofp+1 Lojasiewicz inequalities which express that the multigermof f is stable at any finite set of points outside 0 and onlybecomes unstable at a finite rate when we approach 0. We willdenote this condition by (e). With this notation we prove thefollowing. For any C map germ f:(Rⁿ,0) (R^p,0) the conditions(e), (t), (g') and (a) are equivalent conditions. Moreover,each of these conditions is equivalent to any of (a_k) (p+1 k < , (b_k) (p+1 k < ). 1991 Mathematics Subject Classification:58C27.     

Extremal Properties of Contraction Semigroups on Hilbert and Banach Spaces

Let f be a unit vector and T = {T(t) = e^tA: t 0} be a (C₀)contraction semigroup generated by A on a complex Hilbert spaceX. If |T(t)f,f| 1 as t then f is an eigenvector of A correspondingto a purely imaginary eigenvalue. If one allows X to be a Banachspace, the same situation can be considered by replacing T(t)f,fby (T(t)f) where is a unit vector in X^* dual to f. If |(T(t)f)| 1, as t , is f an eigenvector of A? The answer is sometimesyes and sometimes no.     

On the Reflexivity of the Space {pi}p(E, F) of p-Absolutely Summing Operators, 1 <= p < + {infty}

We give a representation of the dual of the space _p(E,F) ofp-absolutely summing operators (1 p < + ) under certainconditions on E and F. One deduces that the space _p(E, F), 1 p < + , is reflexive if and only if E and F are reflexive.We improve results of Gordon, Lewis, Retherford and Saphar.     

Asymptotics of the m-Coefficient for Eigenvalue Problems with Eigenparameter in the Boundary Conditions

For Sturm-Liouville problems on [a, ) with a regular -dependentboundary condition at a, and the limit point case at , a techniqueof W. N. Everitt [1] is employed to obtain asymptotic formulaefor the associated m()-functions on rays and lines in the complex-plane. The method relies on asymptotic formulae for solutionsof the initial value problem for –u'+qu = u, as || ,which the author has given in [4]. For the case of the regularleft endpoint, the asymptotic formulae on vertical lines sufficeto provide a direct proof of the formula for the total variationof the associated spectral function, a question which the authorhad raised in [3; Remark 5.2].     

Orbits in the Maximal Ideal Space of H{infty}

Let H be the Banach algebra of bounded analytic functions inthe open unit disc D. We can define the rotation in the maximalideal space M(H). For a point x in M(H)\D, an orbit O(x) isnot closed in M(H). It is proved that there exists a point xin M(H) such that x is not contained in the Shilov boundaryX and cl O(x), the closure of O(x), contains X, and there existsa point y in M(H)\(D X) such that cl O(y) X. The rotationpresents many problems concerning H. The purpose of this paperis to discuss these problems.     

Correction: Abelian Varieties defined over their Fields of Moduli, I

Bull London Math. Soc, 4 (1972), 370–372. The proof of the theorem contains an error. Before giving acorrect proof, we state two lemmas. LEMMA 1. Let K/k be a cyclic Galois extension of degree m, let generate Gal (K/k), and let (A, I, ) be defined over K. Supposethat there exists an isomorphism :(A,I,) (A, I, ) over K suchthat v^m–1 ... = 1, where v is the canonical isomorphism(A^m, I^m, ^m) (A, I, ). Then (A, I, ) has a model over k, whichbecomes isomorphic to (A, I, ) over K. Proof. This follows easily from [7], as is essentially explainedon p. 371. LEMMA 2. Let G be an abelian pro-finite group and let : G Q/Z be a continuous character of G whose image has order p.Then either: (a) there exist subgroups G' and H of G such that H is cyclicof order p^m for some m, (G') = 0, and G = G' x H, or (b) for any m > 0 there exists a continuous character m ofG such that p^m _m = . Proof. If (b) is false for a given m, then there exists an element G, of order p^r for some r m, such that () ¦ 0. (Considerthe sequence dual to 0 Ker (p^m) G ^pm G). There exists an opensubgroup G_o of G such that (G₀) = 0 and has order p^r in G/G₀.Choose H to be the subgroup of G generated by , and then aneasy application to G/G₀ of the theory of finite abelian groupsshows the existence of G' (note that () ¦ 0 implies that is not a p-th. power in G). We now prove the theorem. The proof is correct up to the statement(iv) (except that (i) should read: F' k₁ F'ab). To removea minor ambiguity in the proof of (iv), choose to be an elementof Gal (F'_ab/k₂) whose image $$\stackrel{\&macr;}{\sigma}$$ in Gal (k₁/k₂) generates this last group. The error occursin the statement that the canonical map v : A^P A acts on pointsby sending a^p a; it, of course, sends a a. The proof is correct, however, in the case that it is possibleto choose so that ^p = 1 (in Gal (F'/k₂)). By applying Lemma 2 to G = Gal (F'_ab/k₂) and the map G Gal(k₁/k₂) one sees that only the following two cases have to beconsidered. (a) It is possible to choose so that ^pm = 1, for some m, andG = G' x H where G' acts trivially on k₁ and H is generatedby . (b) For any m > 0 there exists a field K, F'ab K k₁ k₂is a cyclic Galois extension of degree p^m. In the first case, we let K F'_ab be the fixed field of G'.Then (A, I, ), regarded as being defined over K, has a modelover k₂. Indeed, if m = 1, then this was observed above, butwhen m > 1 the same argument applies. In the second case, let : (A, I, ) (A$$\stackrel{\&macr;}{\sigma}$$, I$$\stackrel{\&macr;}{\sigma }$$, $$\stackrel{\&macr;}{\sigma}$$) be an isomorphism defined over k₁ and let v ... ^p–1 = µ(R). If is replaced by for some Aut_k1((A, I, )) then is replacedby ^P. Thus, as µ(R) is finite, we may assume that ^pm–1= 1 for some m. Choose K, as in (b), to be of degree p^m overk₂. Let _m be a generator of Gal (K/k₂) whose restriction tok₁ is $$\stackrel{\&macr;}{\sigma }$$. Then : (A, I, ) (A^{$$\stackrel{\&macr;}{\sigma }$$}, I^{$$\stackrel{\&macr;}{\sigma}$$}, ^{$$\stackrel{\&macr;}{\sigma }$$} = (A^{$$\stackrel{\&macr;}{\sigma}$$m}, I^{$$\stackrel{\&macr;}{\sigma }$$m}, ^{$$\stackrel{\&macr;}{\sigma}$$m} is an isomorphism defined over K and v ^mpm–1, ... ^m =^pm–1 = 1, and so, by) Lemma 1, (A, I, ) has a model overk₂ which becomes isomorphic to (A, I, over K. The proof may now be completed as before. Addendum: Professor Shimura has pointed out to me that the claimon lines 25 and 26 of p. 371, viz that µ(R) is a puresubgroup of _R^*t, does not hold for all rings R. Thus this condition,which appears to be essential for the validity of the theorem,should be included in the hypotheses. It holds, for example,if µ(R) is a direct summand of µ(F).     

The l{infty} direct sum of Lp (1 < p < {infty}) is primary

Using Szemeredi's theorem on arithmetic progressions, it isshown that, for 1 < p < , the infinite l direct sum (L^p L^p · · · )_l is a primary Banach space.     

Inequalities for Oscillating p-Functions

Upper bounds on [ ) are derived for those p-functions such thatp() = m is minimum, and p' (+) = q(l-m).     

A reverse Denjoy theorem

Suppose that C₁ and C₂ are two simple curves joining 0 to ,non-intersecting in the finite plane except at 0 and enclosinga domain D which is such that, for all large r, has measure at most 2, where 0 < < .Suppose also that u is a non-constant subharmonic function inthe plane such that u(z) = B(|z|, u) for all large z C₁ C₂.Let A_D(r, u) = inf { u(z):z D and | z | = r }. It is shownthat if A_D(r, u) = O(1) (or A_D(r, u) = o(B(r, u))), then lim_r B(r, u)/r^/2 > 0 (or lim_r log B(r, u)/log r /2).