Standard monomials for <Emphasis Type="Italic">q</Emphasis>-uniform families and a conjecture of Babai and Frankl

Let n, k, α be integers, n, α>0, p be a prime and q=p ^α. Consider the complete q-uniform family

$\mathcal{F}\left( {k,q} \right) = \left\{ {K \subseteq \left[ n \right]:\left| K \right| \equiv k(mod q)} \right\}$

We study certain inclusion matrices attached to F(k,q) over the field\(\mathbb{F}_p \). We show that if l≤q?1 and 2l≤n then

$rank_{\mathbb{F}_p } I(\mathcal{F}(k,q),\left( {\begin{array}{*{20}c} {\left[ n \right]} \\ { \leqslant \ell } \\ \end{array} } \right)) \leqslant \left( {\begin{array}{*{20}c} n \\ \ell \\ \end{array} } \right)$

This extends a theorem of Frankl [7] obtained for the case α=1. In the proof we use arguments involving Gröbner bases, standard monomials and reduction. As an application, we solve a problem of Babai and Frankl related to the size of some L-intersecting families modulo q.     

Fractional Differentiability for Solutions of Nonlinear Elliptic Equations

We study nonlinear elliptic equations in divergence form

$$\text {div }{\mathcal A}(x,Du)=\text {div } G.$$

When \({\mathcal A}\) has linear growth in D u, and assuming that \(x\mapsto {\mathcal A}(x,\xi )\) enjoys \(B^{\alpha }_{\frac {n}\alpha , q}\) smoothness, local well-posedness is found in \(B^{\alpha }_{p,q}\) for certain values of \(p\in [2,\frac {n}{\alpha })\) and \(q\in [1,\infty ]\). In the particular case \({\mathcal A}(x,\xi )=A(x)\xi \), G = 0 and \(A\in B^{\alpha }_{\frac {n}\alpha ,q}\), \(1\leq q\leq \infty \), we obtain \(Du\in B^{\alpha }_{p,q}\) for each \(p<\frac {n}\alpha \). Our main tool in the proof is a more general result, that holds also if \({\mathcal A}\) has growth s?1 in D u, 2 ≤ s ≤ n, and asserts local well-posedness in L ^q for each q > s, provided that \(x\mapsto {\mathcal A}(x,\xi )\) satisfies a locally uniform VMO condition.

     

On exponential sums over primes and application in Waring-Goldbach problem

In this paper, we prove the following estimate on exponential sums over primes: Let κ≥1,βκ=1/2 log κ/log2, x≥2 and α=a/q λsubject to (a, q) = 1, 1≤a≤q, and λ∈R. Then As an application, we prove that with at most O(N2/8 ε) exceptions, all positive integers up to N satisfying some necessary congruence conditions are the sum of three squares of primes. This result is as strong as what has previously been established under the generalized Riemann hypothesis.     

Conflict free colorings of (strongly) almost disjoint set-systems

\(f\: \cup {\mathcal {A}}\to {\rho}\) is called a conflict free coloring of the set-system\({\mathcal {A}}\)(withρcolors) if

$\forall A\in {\mathcal {A}}\ \exists\, {\zeta}<{\rho} (|A\cap f^{-1}\{{\zeta}\}|=1).$

The conflict free chromatic number\(\operatorname {\chi _{\rm CF}}\, ({\mathcal {A}})\) of \({\mathcal {A}}\) is the smallest ρ for which \({\mathcal {A}}\) admits a conflict free coloring with ρ colors.

\({\mathcal {A}}\) is a (λ,κ,μ)-system if \(|{\mathcal {A}}| = \lambda\), |A|=κ for all \(A \in {\mathcal {A}}\), and \({\mathcal {A}}\) is μ-almost disjoint, i.e. |A∩A′|<μ for distinct \(A, A'\in {\mathcal {A}}\). Our aim here is to study

$\operatorname {\chi _{\rm CF}}\, (\lambda,\kappa,\mu) = \sup \{\operatorname {\chi _{\rm CF}}\, ({\mathcal {A}})\: {\mathcal {A}}\mbox{ is a } (\lambda,\kappa,\mu)\mbox{-system}\}$

for λ≧κ≧μ, actually restricting ourselves to λ≧ω and μ≦ω.

For instance, we prove that

? for any limit cardinal κ (or κ=ω) and integers n≧0, k>0, GCH implies

$\operatorname {\chi _{\rm CF}}\, (\kappa^{+n},t,k+1) =\begin{cases}\kappa^{+(n+1-i)}&; \text{if \ } i\cdot k < t \le (i+1)\cdot k,\ i =1,\dots,n;\\[2pt]\kappa&; \text{if \ } (n+1)\cdot k < t;\end{cases}$

? if λ≧κ≧ω>d>1, then λ<κ ^+ω implies \(\operatorname {\chi _{\rm CF}}\, (\lambda,\kappa,d) <\omega\) and λ≧? _ω (κ) implies \(\operatorname {\chi _{\rm CF}}\, (\lambda,\kappa,d) = \omega\);? GCH implies \(\operatorname {\chi _{\rm CF}}\, (\lambda,\kappa,\omega) \le \omega_{2}\) for λ≧κ≧ω ₂ and V=L implies \(\operatorname {\chi _{\rm CF}}\, (\lambda,\kappa,\omega) \le \omega_{1}\) for λ≧κ≧ω ₁;? the existence of a supercompact cardinal implies the consistency of GCH plus \(\operatorname {\chi _{\rm CF}}\,(\aleph_{\omega+1},\omega_{1},\omega)= \aleph_{\omega+1}\) and \(\operatorname {\chi _{\rm CF}}\, (\aleph_{\omega+1},\omega_{n},\omega) = \omega_{2}\) for 2≦n≦ω;? CH implies \(\operatorname {\chi _{\rm CF}}\, (\omega_{1},\omega,\omega) = \operatorname {\chi _{\rm CF}}\, (\omega_{1},\omega_{1},\omega) = \omega_{1}\), while \(MA_{\omega_{1}}\) implies \(\operatorname {\chi _{\rm CF}}\, (\omega_{1},\omega,\omega) = \operatorname {\chi _{\rm CF}}\, (\omega_{1},\omega_{1},\omega) = \omega\).     

The exceptional set for Diophantine inequality with unlike powers of prime variables

Suppose that λ₁, λ₂, λ₃, λ₄ are nonzero real numbers, not all negative, δ > 0, V is a well-spaced set, and the ratio λ₁/λ₂ is algebraic and irrational. Denote by E(V,N, δ) the number of v ∈ V with v ≤ N such that the inequality

$$\left| {{\lambda _1}p_1^2 + {\lambda _2}p_2^3 + {\lambda _3}p_3^4 + {\lambda _4}p_4^5 - \upsilon } \right| < {\upsilon ^{ - \delta }}$$

has no solution in primes p₁, p₂, p₃, p₄. We show that

$$E\left( {\upsilon ,N,\delta } \right) \ll {N^{1 + 2\delta - 1/72 + \varepsilon }}$$

for any ? > 0.

     

A Refinement of the Kolmogorov–Marcinkiewicz–Zygmund Strong Law of Large Numbers

Let {X _n ; n≥1} be a sequence of independent copies of a real-valued random variable X and set S _n =X ₁+???+X _n , n≥1. This paper is devoted to a refinement of the classical Kolmogorov–Marcinkiewicz–Zygmund strong law of large numbers. We show that for 0<p<2,

$\sum_{n=1}^{\infty}\frac{1}{n}\biggl(\frac{|S_{n}|}{n^{1/p}}\biggr)<\infty\quad \mbox{almost surely}$

if and only if

$\begin{cases}\mathbb{E}|X|^{p}<\infty &; \mbox{if }0 < p < 1,\\\mathbb{E}X=0,\ \sum_{n=1}^{\infty}\frac{|\mathbb{E}XI\{|X|\leq n\}|}{n}<\infty,\mbox{ and }\\\sum_{n=1}^{\infty}\frac{\int_{\min\{u_{n},n\}}^{n}\mathbb{P}(|X|>t)\,dt}{n}<\infty &; \mbox{if }p = 1,\\\mathbb{E}X=0\mbox{ and }\int_{0}^{\infty}\mathbb{P}^{1/p}(|X|>t)\,dt<\infty,&;\mbox{if }1 < p < 2,\end{cases}$

where \(u_{n}=\inf \{t:~\mathbb{P}(|X|>t)<\frac{1}{n}\}\), n≥1. Versions of the above result in a Banach space setting are also presented. To establish these results, we invoke the remarkable Hoffmann-Jørgensen (Stud. Math. 52:159–186, 1974) inequality to obtain some general results for sums of the form \(\sum_{n=1}^{\infty}a_{n}\|\sum_{i=1}^{n}V_{i}\|\) (where {V _n ; n≥1} is a sequence of independent Banach-space-valued random variables, and a _n ≥0, n≥1), which may be of independent interest, but which we apply to \(\sum_{n=1}^{\infty}\frac{1}{n}(\frac{|S_{n}|}{n^{1/p}})\).

     

Combinatorial Properties of Rogers-Ramanujan Type Identities Arising from Hall-Littlewood Polynomials

Here we consider the q-series coming from the Hall-Littlewood polynomials,

$$\begin{array}{l}{R_{v} (a, b; q) = {\sum_{\mathop {\lambda}\limits_{\lambda_1 \leq a}}} q^{c | \lambda |} P_{2\lambda}(1, q, q^{2}, \ldots ; q^{2b+d}).}\end{array}$$

These series were defined by Griffin, Ono, and Warnaar in their work on the framework of the Rogers-Ramanujan identities. We devise a recursive method for computing the coefficients of these series when they arise within the Rogers-Ramanujan framework. Furthermore, we study the congruence properties of certain quotients and products of these series, generalizing the famous Ramanujan congruence

$$p(5n+4) \equiv 0\quad ({\rm mod}\, 5).$$

     

Spectral analysis of Darboux transformations for the focusing NLS hierarchy

We study Darboux-type transformations associated with the focusing nonlinear Schrödinger equation (NLS_) and their effect on spectral properties of the underlying Lax operator. The latter is a formallyJ (but nonself-adjoint) Diract-type differential expression of the form

$M(q) = i\left( {\begin{array}{*{20}c} {\frac{d}{{dx}}} &; { - q} \\ { - \bar q} &; { - \frac{d}{{dx}}} \\ \end{array} } \right)$

(1)

satisfying\({\mathcal{J}} M(q)\mathcal{J} = M(q)^* \), whereJ is defined byJ C, andC denotes the antilinear conjugation map in ?²,\({\mathcal{C}}(a,b)^{\rm T} = (\bar a,\bar b)^{\rm T} ,a,b \in \) ?. As one of our principla results, we prove that under the most general hypothesisq ∈ _loc ¹ (?) onq, the maximally defined operatorD(q) generated byM(q) is actually {itJ}-self-adjoint in inL ²(?)². Moreover, we establish the existence of Weyl-Titchmarsh-type solutions ψ+(z, ·) ?L ² ([R, ∞))² and ψ?(z, ·) ∈L ² ((?∞,R]) for allR∈? ofM(q)Ψ ± (z)=zΨ ± (z) forz in the resolvent set ofD(q).

The Darboux transformations considered in this paper are the analogue of the double commutation procedure familiar in the KdV and Schrödinger operator contexts. As in the corresponding case of Schrödinger operators, the Darboux transformations in question guarantee that the resulting potentialsq are locally nonsingular. Moreover, we prove that the construction ofN-soliton NLS_potentialsq ^(N) with respect to a general NLS background potentialq ?L _loc ¹ (?), associated with the Dirac-type operatorsD(q ^(N) ) andD(q), respectively, amounts to the insertio ofN complex conjugate pairs ofL ²({?}²-eigenvalues\(\{ z_1 ,\bar z_1 ,...,z_N ,\bar z_N \} \) into the spectrum σ(D(q)) ofD(q), leaving the rest of the spectrum (especially, the essential spectrum σ_e(itD)(q))) invariant, that is,

$\sigma (D(q^{(N)} )) = \sigma (D(q)) \cup \{ z_1 ,\bar z_1 ,...,z_N ,\bar z_N \} ,$

(1)

$\sigma _e (D(q^{(N)} )) = \sigma _e (D(q))$

(1)

These results are obtained by establishing the existence of bounded transformation operators which intertwine the background Dirac operatorD(q) and the Dirac operatorD(q ^(N) ) obtained afterN Darboux-type transformations.     

Classification of certain qualitative properties of solutions for the quasilinear parabolic equations

In this paper, we mainly consider the initial boundary problem for a quasilinear parabolic equation u_t-div(|?u|~(p-2)?u) =-|u|~(β-1) u + α|u|~(q-2 )u,where p 1, β 0, q≥1 and α 0. By using Gagliardo-Nirenberg type inequality, the energy method and comparison principle, the phenomena of blowup and extinction are classified completely in the different ranges of reaction exponents.     

Entire solutions of Fermat type differential-difference equations

We mainly discuss entire solutions with finite order of the following Fermat type differential-difference equations

$$\begin{array}{ll}(f)^{n}+f(z+c)^{m}=1;\\f^{\prime}(z)^{n}+f(z+c)^{m}=1;\\ f^{\prime}(z)^{n}+[f(z+c)-f(z)]^{m}=1,\end{array}$$

where m, n are positive integers.

     

Convergence rates for Tikhonov regularization of a two-coefficient identification problem in an elliptic boundary value problem

We investigate the convergence rates for Tikhonov regularization of the problem of simultaneously estimating the coefficients q and a in the Neumann problem for the elliptic equation \({{-{\rm div}(q \nabla u) + au = f \;{\rm in}\; \Omega, q{\partial u}/{\partial n} = g}}\) on the boundary \({{\partial\Omega, \Omega \subset \mathbb{R}^d, d \geq 1}}\) , when u is imprecisely given by \({{{z^\delta} \in {H^1}(\Omega), \|u-z^\delta\|_{H^1(\Omega)}\le\delta, \delta > 0}}\). We regularize this problem by minimizing the strictly convex functional of (q, a)

$\begin{array}{lll}\int\limits_{\Omega}\left(q| \nabla (U(q,a)-z^{\delta})|^2 + a(U(q,a)-z^{\delta})^2\right)dx\\\quad+\rho\left(\|q-q^*\|^2_{L^2(\Omega)} + \|a-a^*\|^2_{L^2(\Omega)}\right)\end{array}$

over the admissible set K, where ρ > 0 is the regularization parameter and (q*, a*) is an a priori estimate of the true pair (q, a) which is identified, and consider the unique solution of these minimization problem as the regularized one to that of the inverse problem. We obtain the convergence rate \({{{\mathcal {O}}(\sqrt{\delta})}}\), as δ → 0 and ρ ~ δ, for the regularized solutions under the simple and weak source condition

${\rm there\;is\;a\;function}\;w^* \in V^*\;{\rm such\;that}\;{U^\prime (q^ \dagger, a^\dagger)}^*w^* = (q^\dagger - q^*, a^\dagger - a^*)$

with \({{(q^\dagger, a^\dagger)}}\) being the (q*, a*)-minimum norm solution of the coefficient identification problem, U′(·, ·) the Fréchet derivative of U(·, ·), V the Sobolev space on which the boundary value problem is considered. Our source condition is without the smallness requirement on the source function which is popularized in the theory of regularization of nonlinear ill-posed problems. Furthermore, some concrete cases of our source condition are proved to be simply the requirement that the sought coefficients belong to certain smooth function spaces.

     

A remark on the values of binary linear forms at prime arguments

Let λ₁, λ₂ be positive real numbers such that \({\frac{{\lambda_1}}{{\lambda_2}}}\) is irrational and algebraic. For any (C, c) well-spaced sequence \({\mathcal {V} = \{{v_i}\}_{i = 1}^\infty}\) and δ > 0 let \({E( {\mathcal {V},X,\delta})}\) denote the number of elements \({v \in \mathcal {V}, v \le X}\) for which the inequality

$| {\lambda_1 p_1 + \lambda_2 p_2 - v} | < X^{- \delta}$

is not solvable in primes p ₁, p ₂. In this paper it is proved that

$E( {\mathcal {V},X,\delta}) \ll X^{\frac{4}{5} + \delta + \varepsilon}$

for any \({\varepsilon > 0}\). This result constitutes an improvement upon that of Brüdern, Cook, and Perelli for the range \({\frac{2}{{15}} < \delta < \frac{1}{5}}\).

     

Reconstructing a matrix from a partial sampling of Pareto eigenvalues

Let Λ={λ ₁,…,λ _p } be a given set of distinct real numbers. This work deals with the problem of constructing a real matrix A of order n such that each element of Λ is a Pareto eigenvalue of A, that is to say, for all k∈{1,…,p} the complementarity system

$x\geq \mathbf{0}_n,\quad Ax-\lambda_k x\geq \mathbf{0}_n,\quad \langle x, Ax-\lambda_k x\rangle = 0$

admits a nonzero solution x∈? ⁿ .

     

Local boundedness of variational solutions to evolutionary problems with non-standard growth

We prove the local boundedness of variational solutions and parabolic minimizers to evolutionary problems, where the integrand f is convex and satisfies a non-standard p, q-growth condition with

$$1 < p \leq q \leq p \tfrac{n+2}{n}.$$

A function \({u\colon \Omega_T := \Omega \times (0,T) \to \mathbb{R}}\) is called parabolic minimizer if it satisfies the minimality condition

$$\int_{\Omega_T} u \cdot \partial_t \varphi +f(x, Du) {\rm d} z \leq \int_{\Omega_T} f(x, Du + D \varphi) {\rm d}z$$

for every \({\varphi \in C^\infty_0(\Omega_T)}\). Moreover, we will show local boundedness for parabolic minimizers, if f satisfies an anisotropic growth condition.

     

Mean Square Estimates for Coefficients of Symmetric Power <Emphasis Type="Italic">L</Emphasis>-Functions

Let L(sym ^j f,s) be the jth symmetric power L-function attached to a holomorphic Hecke eigencuspform f(z) for the full modular group, and \(\lambda_{\mathrm{sym}^{j}f}(n)\) denote its nth coefficient. In this paper we are able to prove that

$\int_{1}^{x}\bigg|\sum_{n\leq y}\lambda_{\mathrm{sym}^{3}f}(n)\bigg|^{2}dy=O\bigl(x^{2}\bigr),$

and

$\int_{1}^{x}\bigg|\sum_{n\leq y}\lambda_{\mathrm{sym}^{4}f}(n)\bigg|^{2}dy=O\bigl(x^{\frac{11}{5}}\log x\bigr).$

     

Harnack Inequality for Non-Local Schrödinger Operators

Let \(x \in \mathbb {R}^{d}\), d ≥ 3, and \(f: \mathbb {R}^{d} \rightarrow \mathbb {R}\) be a twice differentiable function with all second partial derivatives being continuous. For 1 ≤ i, j ≤ d, let \(a_{ij} : \mathbb {R}^{d} \rightarrow \mathbb {R}\) be a differentiable function with all partial derivatives being continuous and bounded. We shall consider the Schrödinger operator associated to

$$\mathcal{L}f(x) = \frac12 \sum\limits_{i=1}^{d} \sum\limits_{j=1}^{d} \frac{\partial}{\partial x_{i}} \left( a_{ij}(\cdot) \frac{\partial f}{\partial x_{j}}\right)(x) + {\int}_{\mathbb{R}^{d}\setminus{\{0\}}} [f(y) - f(x) ]J(x,y)dy $$

where \(J: \mathbb {R}^{d} \times \mathbb {R}^{d} \rightarrow \mathbb {R}\) is a symmetric measurable function. Let \(q: \mathbb {R}^{d} \rightarrow \mathbb {R}.\) We specify assumptions on a, q, and J so that non-negative bounded solutions to

$$\mathcal{L}f + qf = 0 $$

satisfy a Harnack inequality. As tools we also prove a Carleson estimate, a uniform Boundary Harnack Principle and a 3G inequality for solutions to \(\mathcal {L}f = 0.\)

     

Univoque bases and Hausdorff dimension

Given a positive integer M and a real number \(q >1\), a q -expansion of a real number x is a sequence \((c_i)=c_1c_2\ldots \) with \((c_i) \in \{0,\ldots ,M\}^\infty \) such that

$$\begin{aligned} x=\sum _{i=1}^{\infty } c_iq^{-i}. \end{aligned}$$

It is well known that if \(q \in (1,M+1]\), then each \(x \in I_q:=\left[ 0,M/(q-1)\right] \) has a q-expansion. Let \(\mathcal {U}=\mathcal {U}(M)\) be the set of univoque bases \(q>1\) for which 1 has a unique q-expansion. The main object of this paper is to provide new characterizations of \(\mathcal {U}\) and to show that the Hausdorff dimension of the set of numbers \(x \in I_q\) with a unique q-expansion changes the most if q “crosses” a univoque base. Denote by \(\mathcal {B}_2=\mathcal {B}_2(M)\) the set of \(q \in (1,M+1]\) such that there exist numbers having precisely two distinct q-expansions. As a by-product of our results, we obtain an answer to a question of Sidorov (J Number Theory 129:741–754, 2009) and prove that

$$\begin{aligned} \dim _H(\mathcal {B}_2\cap (q',q'+\delta ))>0\quad \text {for any}\quad \delta >0, \end{aligned}$$

where \(q'=q'(M)\) is the Komornik–Loreti constant.

     

On the existence of periodic motions of the excited inverted pendulum by elementary methods

Using purely elementary methods, necessary and sufficient conditions are given for the existence of 2T-periodic and 4T-periodic solutions around the upper equilibrium of the mathematical pendulum when the suspension point is vibrating with period 2T. The equation of the motion is of the form

$$\ddot{\theta}-\frac{1}{l}(g+a(t)) \theta=0,$$

where l, g are constants and

$$a(t) := \begin{cases} A &\text{if } 2kT\leq t < (2k+1)T,\\ -A &\text{if } (2k+1)T\leq t < (2k+2)T,\end{cases}\quad (k=0,1,\dots);$$

A, T are positive constants. The exact stability zones for the upper equilibrium are presented.

     

Numerical Radius Inequalities for Certain 2 × 2 Operator Matrices

We prove several numerical radius inequalities for certain 2 × 2 operator matrices. Among other inequalities, it is shown that if X, Y, Z, and W are bounded linear operators on a Hilbert space, then

$$w\left( \left[\begin{array}{cc} X &; Y \\ Z &; W \end{array} \right] \right) \geq \max \left(w(X),w(W),\frac{w(Y+Z)}{2},\frac{w(Y-Z)}{2}\right) $$

and

$$w\left( \left[\begin{array}{cc}X &; Y \\ Z &; W\end{array} \right] \right) \leq \max \left( w(X), w(W)\right)+\frac{w(Y+Z)+w(Y-Z)}{2}. $$

As an application of a special case of the second inequality, it is shown that

$$\frac{\left\Vert X\right\Vert }{2}+\frac{\left\vert \left\Vert\operatorname{Re}{X}\right\Vert -\frac{\left\Vert X\right\Vert}{2}\right\vert }{4}+\frac{ \left\vert \left\Vert \operatorname{Im}{X}\right\Vert -\frac{\left\Vert X\right\Vert}{2}\right\vert }{4} \leq w(X), $$

which is a considerable improvement of the classical inequality \({\frac{ \left\Vert X\right\Vert }{2}\leq w(X)}\) . Here w(·) and || · || are the numerical radius and the usual operator norm, respectively.

     

Two-weight inequality for operator-valued positive dyadic operators by parallel stopping cubes

We study the operator-valued positive dyadic operator

$${T_\lambda }\left( {f\sigma } \right): = \sum\limits_{Q \in D} {{\lambda _Q}} \int_Q {fd\sigma 1Q}, $$

where the coefficients {λ _Q : C → D} _Q∈D are positive operators from a Banach lattice C to a Banach lattice D. We assume that the Banach lattices C and D* each have the Hardy–Littlewood property. An example of a Banach lattice with the Hardy–Littlewood property is a Lebesgue space.

In the two-weight case, we prove that the L _C ^p (σ) → L _D ^q (ω) boundedness of the operator T _λ( · σ) is characterized by the direct and the dual L ^∞ testing conditions:

$$\left\| {{1_Q}{T_\lambda }} \right\|{\left( {{1_Q}f\sigma } \right)||_{L_D^q\left( \omega \right)}} \lesssim {\left\| f \right\|_{L_C^\infty \left( {Q,\sigma } \right)}}\sigma {\left( Q \right)^{1/p}}$$

,

$${\left\| {{1_Q}{T_\lambda }*\left( {{1_{Qg\omega }}} \right)} \right\|_{L_{C*}^{p'}\left( \sigma \right)}} \lesssim {\left\| g \right\|_{L_{D*}^\infty \left( {Q,\omega } \right)}}\omega {\left( Q \right)^{1/q'}}$$

.

Here L _C ^p (σ) and L _D ^q (ω) denote the Lebesgue–Bochner spaces associated with exponents 1 < p ≤ q < ∞, and locally finite Borel measures σ and ω.

In the unweighted case, we show that the L _C ^p (μ) → L _D ^p (μ) boundedness of the operator T _λ( · μ) is equivalent to the end-point direct L ^∞ testing condition:

$${\left\| {{1_Q}{T_\lambda }\left( {{1_Q}f\mu } \right)} \right\|_{L_D^1\left( \mu \right)}} \lesssim {\left\| f \right\|_{L_C^\infty \left( {Q,\mu } \right)}}\left( {Q,\mu } \right)\mu \left( Q \right)$$

.

This condition is manifestly independent of the exponent p. By specializing this to particular cases, we recover some earlier results in a unified way.