Local atomic decompositions for multidimensional Hardy spaces

<jats:p>We consider a nonnegative self-adjoint operator <jats:italic>L</jats:italic> on <jats:inline-formula><jats:alternatives><jats:tex-math>$$L^2(X)$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
                  <mml:mrow>
                    <mml:msup>
                      <mml:mi>L</mml:mi>
                      <mml:mn>2</mml:mn>
                    </mml:msup>
                    <mml:mrow>
                      <mml:mo>(</mml:mo>
                      <mml:mi>X</mml:mi>
                      <mml:mo>)</mml:mo>
                    </mml:mrow>
                  </mml:mrow>
                </mml:math></jats:alternatives></jats:inline-formula>, where <jats:inline-formula><jats:alternatives><jats:tex-math>$$X\subseteq {{\mathbb {R}}}^d$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
                  <mml:mrow>
                    <mml:mi>X</mml:mi>
                    <mml:mo>⊆</mml:mo>
                    <mml:msup>
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                        <mml:mi>R</mml:mi>
                      </mml:mrow>
                      <mml:mi>d</mml:mi>
                    </mml:msup>
                  </mml:mrow>
                </mml:math></jats:alternatives></jats:inline-formula>. Under certain assumptions, we prove atomic characterizations of the Hardy space <jats:disp-formula><jats:alternatives><jats:tex-math>$$\begin{aligned} H^1(L) = \left\{ f\in L^1(X) \ : \ \left\| \sup _{t>0} \left| \exp (-tL)f \right| \right\| _{L^1(X)}<\infty \right\} . \end{aligned}$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
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                              <mml:mo>(</mml:mo>
                              <mml:mi>L</mml:mi>
                              <mml:mo>)</mml:mo>
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                                <mml:mi>L</mml:mi>
                                <mml:mn>1</mml:mn>
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                                      <mml:mn>0</mml:mn>
                                    </mml:mrow>
                                  </mml:munder>
                                  <mml:mfenced>
                                    <mml:mo>exp</mml:mo>
                                    <mml:mo>(</mml:mo>
                                    <mml:mo>-</mml:mo>
                                    <mml:mi>t</mml:mi>
                                    <mml:mi>L</mml:mi>
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                                  </mml:mfenced>
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                                    <mml:mi>L</mml:mi>
                                    <mml:mn>1</mml:mn>
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                                    <mml:mo>(</mml:mo>
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                              <mml:mo><</mml:mo>
                              <mml:mi>∞</mml:mi>
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                      </mml:mtr>
                    </mml:mtable>
                  </mml:mrow>
                </mml:math></jats:alternatives></jats:disp-formula>We state simple conditions, such that <jats:inline-formula><jats:alternatives><jats:tex-math>$$H^1(L)$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
                  <mml:mrow>
                    <mml:msup>
                      <mml:mi>H</mml:mi>
                      <mml:mn>1</mml:mn>
                    </mml:msup>
                    <mml:mrow>
                      <mml:mo>(</mml:mo>
                      <mml:mi>L</mml:mi>
                      <mml:mo>)</mml:mo>
                    </mml:mrow>
                  </mml:mrow>
                </mml:math></jats:alternatives></jats:inline-formula> is characterized by atoms being either the classical atoms on <jats:inline-formula><jats:alternatives><jats:tex-math>$$X\subseteq {\mathbb {R}^d}$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
                  <mml:mrow>
                    <mml:mi>X</mml:mi>
                    <mml:mo>⊆</mml:mo>
                    <mml:msup>
                      <mml:mrow>
                        <mml:mi>R</mml:mi>
                      </mml:mrow>
                      <mml:mi>d</mml:mi>
                    </mml:msup>
                  </mml:mrow>
                </mml:math></jats:alternatives></jats:inline-formula> or local atoms of the form <jats:inline-formula><jats:alternatives><jats:tex-math>$$|Q|^{-1}\chi _Q$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
                  <mml:mrow>
                    <mml:msup>
                      <mml:mrow>
                        <mml:mo>|</mml:mo>
                        <mml:mi>Q</mml:mi>
                        <mml:mo>|</mml:mo>
                      </mml:mrow>
                      <mml:mrow>
                        <mml:mo>-</mml:mo>
                        <mml:mn>1</mml:mn>
                      </mml:mrow>
                    </mml:msup>
                    <mml:msub>
                      <mml:mi>χ</mml:mi>
                      <mml:mi>Q</mml:mi>
                    </mml:msub>
                  </mml:mrow>
                </mml:math></jats:alternatives></jats:inline-formula>, where <jats:inline-formula><jats:alternatives><jats:tex-math>$$Q\subseteq X$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
                  <mml:mrow>
                    <mml:mi>Q</mml:mi>
                    <mml:mo>⊆</mml:mo>
                    <mml:mi>X</mml:mi>
                  </mml:mrow>
                </mml:math></jats:alternatives></jats:inline-formula> is a cube (or cuboid). One of our main motivation is to study multidimensional operators related to orthogonal expansions. We prove that if two operators <jats:inline-formula><jats:alternatives><jats:tex-math>$$L_1, L_2$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
                  <mml:mrow>
                    <mml:msub>
                      <mml:mi>L</mml:mi>
                      <mml:mn>1</mml:mn>
                    </mml:msub>
                    <mml:mo>,</mml:mo>
                    <mml:msub>
                      <mml:mi>L</mml:mi>
                      <mml:mn>2</mml:mn>
                    </mml:msub>
                  </mml:mrow>
                </mml:math></jats:alternatives></jats:inline-formula> satisfy the assumptions of our theorem, then the sum <jats:inline-formula><jats:alternatives><jats:tex-math>$$L_1 + L_2$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
                  <mml:mrow>
                    <mml:msub>
                      <mml:mi>L</mml:mi>
                      <mml:mn>1</mml:mn>
                    </mml:msub>
                    <mml:mo>+</mml:mo>
                    <mml:msub>
                      <mml:mi>L</mml:mi>
                      <mml:mn>2</mml:mn>
                    </mml:msub>
                  </mml:mrow>
                </mml:math></jats:alternatives></jats:inline-formula> also does. As a consequence, we give atomic characterizations for multidimensional Bessel, Laguerre, and Schrödinger operators. As a by-product, under the same assumptions, we characterize <jats:inline-formula><jats:alternatives><jats:tex-math>$$H^1(L)$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
                  <mml:mrow>
                    <mml:msup>
                      <mml:mi>H</mml:mi>
                      <mml:mn>1</mml:mn>
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                    <mml:mrow>
                      <mml:mo>(</mml:mo>
                      <mml:mi>L</mml:mi>
                      <mml:mo>)</mml:mo>
                    </mml:mrow>
                  </mml:mrow>
                </mml:math></jats:alternatives></jats:inline-formula> also by the maximal operator related to the subordinate semigroup <jats:inline-formula><jats:alternatives><jats:tex-math>$$\exp (-tL^\nu )$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
                  <mml:mrow>
                    <mml:mo>exp</mml:mo>
                    <mml:mo>(</mml:mo>
                    <mml:mo>-</mml:mo>
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                      <mml:mi>L</mml:mi>
                      <mml:mi>ν</mml:mi>
                    </mml:msup>
                    <mml:mo>)</mml:mo>
                  </mml:mrow>
                </mml:math></jats:alternatives></jats:inline-formula>, where <jats:inline-formula><jats:alternatives><jats:tex-math>$$\nu \in (0,1)$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
                  <mml:mrow>
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                  </mml:mrow>
                </mml:math></jats:alternatives></jats:inline-formula>.
</jats:p>


Introduction
Let us first recall that the classical Hardy space H 1 (R d ) can be defined by the maximal operator, i.e.
Here and thereafter H t = exp(t ) is the heat semigroup on R d given by (1.1) Among many equivalent characterizations of H 1 (R d ) one of the most useful is the characterization by atomic decompositions proved by Coifman [4] in the onedimensional case and by Latter [19] in the general case d ∈ N. It says that f ∈ H 1 (R d ) if and only if f (x) = ∞ k=1 λ k a k (x), where λ k ∈ C are such that ∞ k=1 |λ k | < ∞ and a k are atoms. By definition, a function a is an atom if there exists a ball B ⊆ R d such that: i.e. a satisfies well-known localization, size, and cancellation conditions. Later, Goldberg in [16] noticed that if we restrict the supremum in the maximal operator above to the range t ∈ (0, τ 2 ), with τ > 0 fixed, then still the atomic characterization holds, but with additional atoms of the form a(x) = |B| −1 χ B (x), where χ is the characteristic function and B is a ball of radius τ (see Sect. 2 for details).
In this paper we deal with atomic characterizations of the Hardy space H 1 for operators, such that H 1 admits atoms of local type, i.e. atoms of the form |B| −1 χ B . We shall consider operators defined on L 2 (X ), where X ⊆ R d with the Lebesgue measure. Our main focus will be on sums of the form L = L 1 + · · · + L d , where each L i acts only on the variable x i , where x = (x 1 , ..., x d ). For such L we look for atomic decompositions. As an application, we can take operators related to some multidimensional orthogonal expansions. Additionally we prove characterizations of H 1 by subordinate semigroups.

Notation
Let X = (a 1 , b 1 )×· · ·×(a d , b d ) be a subset of R d . We allow a j = −∞ and b j = ∞ so that we consider products of lines, half-lines, and finite intervals. We equip X with the Euclidean metric and the Lebesgue measure. In the product case it is more convenient to use cubes and cuboids instead of balls, so denote for z = (z 1 , ..., z d ) ∈ X and r 1 , ..., r d > 0 the closed cuboid and the cube Q(z, r ) = Q(z, r , ..., r ). We shall call such z the center of a cube/cuboid. For a cuboid Q by d Q we shall denote the diameter of Q. Definition 1.2 Let Q be a set of cuboids in X . We call Q an admissible covering of X if there exist C 1 , C 2 > 0 such that: Let us note that 3. means that our cuboids are almost cubes. In fact, we shall often use only cubes.
By Q * we shall denote a slight enlargement of Q. More precisely, if Q = (z, r 1 , ..., r d ), then Q * := Q(z, κr 1 , ..., κr d ), where κ > 1. Observe that if Q is an admissible covering of R d , then choosing κ close enough to 1 the family {Q * * * } Q∈Q is a finite covering of R d , namely and, for Q 1 , Q 2 ∈ Q, In this paper we always choose κ such that (1.3) and (1.4) are satisfied. Let us emphasize that Q and Q * are always defined as a subset of X , not as a subset of R d . Having two admissible coverings Q 1 and Q 2 on R d 1 and R d 2 we would like to produce an admissible covering on R d 1 +d 2 . However, one simply observe that products {Q 1 × Q 2 : Q 1 ∈ Q 1 , Q 2 ∈ Q 2 }, would not produce admissible covering (in general, 3. would fail). Therefore, for the sake of this paper, let us state the following definition.
Definition 1.5 Assume that Q 1 and Q 2 are admissible coverings of X 1 ⊆ R d 1 and X 2 ⊆ R d 2 , respectively. We define an admissible covering of X 1 × X 2 in the following way. First, consider the covering {Q 1 × Q 2 : Q 1 ∈ Q 1 , Q 2 ∈ Q 2 }. Then we further split each Q = Q 1 × Q 2 . Without loss of generality let us assume that d Q 1 > d Q 2 .
We split Q 1 into cuboids Q We shall denote such covering by Q 1 Q 2 . One may check that the definition above leads to an admissible covering of X 1 × X 2 .
Having an admissible covering Q of X ⊆ R d we define a local atomic Hardy space H 1 at (Q) related to Q in the following way. We say that a function a : X → C is a Q − atom if: (i) either there is Q ∈ Q and a cube K ⊂ Q * , such that: Having Q-atoms we define the local atomic Hardy space related to Q, H 1 at (Q), in a standard way. Namely, we say that a function f is in with k |λ k | < ∞ and a k being Q-atoms. Moreover, the norm of H 1 at (Q) is given by where the infimum is taken over all possible representations of f (x) = k λ k a k (x) as above. One may simply check that H 1 at (Q) is a Banach space. In the whole paper by L we shall denote a nonnegative self-adjoint operator and by T t = exp(−t L) the heat semigroup generated by L. We shall always assume that there exists a nonnegative integral kernel Our initial definition of the Hardy space H 1 (L) shall be given by means of the maximal operator associated with T t , namely Moreover, we shall consider the subordinate semigroup K t,ν = exp(−t L ν ), ν ∈ (0, 1), and its Hardy space, which is defined by

Main results
Let us assume that an admissible covering Q of X is given. Recall that H t (x, y) is the classical semigroup on R d given in (1.1), and denote by P t,ν = exp(−t(− ) ν ) the semigroup generated by (− ) ν , ν ∈ (0, 1), and given by P t,ν f (x) = R d P t,ν (x, y) f (y) dy. The kernel P t,ν (x, y) is a transition density of the symmetric 2ν-stable Lévy process in R d . It is well-known that see e.g. [18,Subsec. 2.6], [15]. Let us mention that in the particular case of ν = 1/2, the semigroup P t,1/2 is the well-known Poisson semigroup on R d . Assume that an operator L is as in Sect. 1.2. Let ν ∈ (0, 1) and suppose that T t (x, y) is either H t (x, y) or P t ν ,ν (x, y). Consider the following assumptions: Theorem A Assume that for L, T t , and an admissible covering Q the conditions (A 0 )-(A 2 ) hold. Then H 1 (L) = H 1 at (Q) and the corresponding norms are equivalent.
The proof of Theorem A is standard and uses only local characterization of Hardy spaces as in [16]. For the convenience of the reader we present the proof in Sect. 3.
Our first main goal is to describe atomic characterizations for sums of the form L 1 + · · · + L N , where each L j satisfies ( A 0 )-(A 2 ) on a proper subspace. This is very useful in many cases such as multidimensional orthogonal expansions. Instead of dealing with products of kernels of semigroups, we can consider only one-dimensional kernel, but we shall need to prove slightly stronger conditions. More precisely, we consider Assume that L i is an operator on L 2 (X i ), as in Sect. 1.2. Slightly abusing the notation we keep the symbol L i for I ⊗ ... ⊗ L i ⊗ ... ⊗ I as the operator on L 2 (X ) and denote (1.7) .., N , is nonnegative and has the upper Gaussian estimates, namely t (x N , y N ). Moreover, we shall assume that for each i ∈ {1, ..., N } there exist a proper covering Q i of R d i such that the following generalizations of ( A 1 ) and (A 2 ) hold: there exists γ ∈ (0, 1/3) such that for every δ ∈ [0, γ ) and every i = 1, .., N , Here H t is the classical heat semigroup on R d i , depending on the context. Now, we are ready to state our first main theorem.
Theorem B Assume that for i = 1, ..., N kernels T [i] t (x i , y i ) are related to L i and suppose that for T [i] t (x i , y i ) together with admissible coverings Q i the conditions (A 0 )-(A 2 ) hold. If L = L 1 + · · · + L N is as in (1.7), then and the corresponding norms are equivalent.
Our second main goal is to characterize H 1 (L) by the subordinate semigroup K t,ν = exp(−t L ν ), for 0 < ν < 1. Obviously, one can try to apply Theorem A, but for many operators the subordinate kernel K t,ν (x, y) is harder to analyze than T t (x, y) (e.g., in some cases a concrete formula with special functions exists for T t (x, y), but not for K t,ν (x, y)). However, it appears that under our assumptions (A 0 )-(A 2 ) we obtain the characterization by the subordinate semigroup essentially for free.
Theorem C Under the assumptions of Theorem B, for ν ∈ (0, 1), we have that Moreover, the corresponding norms are equivalent.

Applications
One of the goals of this paper is to verify the assumptions of Theorems B and C for various well-known operators. In this subsection we provide a list of such operators.

Bessel operator
For β > 0 let L Here, I τ is the modified Bessel function of the first kind. The Hardy space B ) for the one-dimensional Bessel operator was studied in [2]. In Sect. 4.1 we check that the assumptions (A 0 )-(A 2 ) are satisfied for L B with the admissible covering  (Fig. 1). Moreover, the associated norms are comparable.

Laguerre operator
denote the Laguerre operator on L was studied in [7]. The admissible covering is the following Fig. 2 for Q l Q L . Using methods similar to those in [7] we verify (A 0 )-(A 2 ) in Sect. 4.2.
Moreover, the associated norms are comparable.

Schrödinger operators
where T S,t = exp(−t L S ) and H t = exp(t ), see (1.1). Following [11], for fixed V , we assume that there is an admissible covering Q S of R d that satisfies the following conditions: there exist constants ρ > 1 and σ > 0 such that The Hardy spaces related to Schrödinger operators have been widely studied. It appears that for some potentials the atoms for H 1 (L S ) have local nature (as in our paper), but this is no longer true for other potentials. The interested reader is referred to [5,8,9,[11][12][13][14]17].
In [11] the authors study potentials as above, but instead of assuming (D') they have a bit more general assumption (D), which instead of ρ −n has an arbitrary summable sequence (1 + n) −1−ε on the right-hand side of (D'). Moreover, the assumptions (D') and (K) are easy to generalize for products, see [8,Rem. 1.8]. Therefore, for Schrödinger operators Theorem B is a bit weaker than results of [11]. However, Theorem C gives additionally characterization by the semigroups exp −t L ν S , 0 < ν < 1, provided that the stronger assumption (D') is satisfied. Let us notice that indeed (D') is true for many examples, including L S in dimension one with any nonnegative V ∈ L 1 loc (R), see [5]. In Sect. 4.2 we prove that (D') and (K) imply the assumptions of Theorems B and C , which leads to the following.

Product of local and nonlocal atomic Hardy space
As we have mentioned, all atoms on the Hardy space H 1 (R d 1 ) satisfy cancellation condition, i.e. they are nonlocal atoms. However, if we consider the product R d = R d 1 × R d 2 and the operator L = − + L 2 , where L 2 and Q 2 satisfies the assumptions (A 0 )-(A 2 ) on R d 2 then the resulting Hardy space H 1 (L) shall have local character.
More precisely, if R d 1 Q 2 is the admissible covering that arise by splitting all the strips R d 1 × Q 2 , Q 2 ∈ Q 2 , into countable many cuboids Q 1,n × Q 2 , where Q 1,n = Q(z n , d Q 2 ). Then we have the following corollary (see Sect. 4.4).

Organization of the paper
The paper is organized in the following way. Section 2 is devoted to prove some preliminary estimates and to recall some known facts about local Hardy spaces on R d . In Sect. 3 we prove our main results, namely Theorems A, B, and C . In Sect. 4 we prove that the examples given in Sect. 1.4 satisfy assumptions (A 0 )-(A 2 ). We use standard notation, i.e. C denotes some constant that can change from line to line.

Auxiliary estimates
For an admissible covering Q of X let us denote for Q ∈ Q the functions ψ Q ∈ C 1 (X ) satisfying It is easy to observe that such family ψ Q Q∈Q exists, provided that Q satisfies Definition 1.3. The family ψ Q Q∈Q shall be called a partition of unity related to Q.

Proposition 2.2 Assume that T t , and an admissible covering Q satisfy (A 0 ) and (A 1 ).
Let ψ Q be a partition of unity related to Q. Then

3)
and We now turn to prove (2.4). Fix y ∈ X and Q 0 ∈ Q such that y ∈ Q 0 . Denote N (Q 0 ) = Q ∈ Q : Q * * * 0 ∩ Q * * * = ∅ (the neighbors of Q 0 ) . Notice that |N (Q 0 )| ≤ C, see (1.3). Then Notice that for Q ∈ N (Q 0 ) we have d Q d Q 0 . To deal with S 1 we use (A 0 ) and the mean value theorem for ψ Q , To estimate S 2 we use ψ Q ∞ ≤ 1 and (A 1 ), getting

Lemma 2.5 Assume that T t satisfy (A 0 ).
Then, for f ∈ L 1 (X ) + L ∞ (X ), The proof of the Lemma 2.5 goes by standard arguments. For the convenience of the reader we present details in Appendix.

Local Hardy spaces
In this section, we recall some classical results on local Hardy spaces, see [16]. Let τ > 0 be fixed. We are interested in decomposing into atoms a function f such that (2.6) It is known, that (2.6) holds if and only if f (x) = k λ k a k (x), where k |λ k | < ∞ and a k are either the classical atoms or the local atoms at scale τ . The latter are atoms a supported in a cube Q of diameter at most τ such that a ∞ ≤ |Q| −1 but we do not impose the cancellation condition. In other words one may say that this is the space H 1 at (Q {τ } ) introduced in Sect. 1.2, where Q {τ } is a covering of R d by cubes with diameter τ . The next proposition states the local atomic decomoposition theorem in a version that will be suitable for us in the proof of Theorem A. This proposition can be obtained by known methods from the global characterization of the classical Hardy space H 1 (R d ). One may also check the assumptions from a general result of Uchiyama [23,Cor. 1']. The details are left for the interested reader.

Proposition 2.7
Let τ > 0 be fixed and T t denote either H t or P t ν ,ν , see (1.1) and (1.6). Then, there exists C > 0 that does not depend on τ such that:

For every classical atom a or an atom of the form a(x)
then there exist sequences {λ k } k and {a k (x)} k , such that f (x) = k λ k a k (x), k |λ k | ≤ C M, and a k are either the classical atoms supported in Q * or a k (x) = |Q| −1 χ Q (x). Remark 2.8 Proposition 2.7 remains valid for many other kernels T t satisfying (A 0 ) and, therefore, Theorem A holds for such kernels.

Proof of Theorem A
Proof Recall that by the assumptions and Proposition 2.2 we also have that (2.3) and (2.4) are satisfied. We shall prove two inclusions.
First inequality: . It suffices to show that for every Qatom a we have sup t>0 |T t a| L 1 (X ) ≤ C, where C does not depend on a. Let a be associated with a cuboid Q ∈ Q, i.e. supp a ⊂ Q * . Recall that T t is either H t or P t ν ,ν , see (1.1) and (1.6). Observe that by using (A 1 ), (A 2 ), (2.3), and part 1. of Proposition 2.7 we get sup t>0 |T t a| Let ψ Q be a partition of unity related to Q, see (2.1). We have f = Q∈Q ψ Q f . Denote f Q = ψ Q f and notice that since supp f Q ⊂ Q * , then Using (3.1)-(3.4) and Lemma 2.5 we arrive at . Now, from part 2. of Proposition 2.7 for each f Q we obtain λ Q,k , a Q,k . Then .
Finally, we notice that all the atoms a Q,k obtained by Proposition 2.7 are indeed Q-atoms.

Remark 3.5
The assumption (A 0 ) has only been used in Proposition 2.2. Therefore, in Theorem A one may replace the assumption ( A 0 ) by the pair of assumptions (2.3) and (2.4).

Proof of Theorem B
Proof We shall show the following claim. If the assumptions ( A 0 )-(A 2 ) hold for T [ j] t (x j , y j ) together with admissible coverings Q j for j = 1, 2, then ( A 0 )-(A 2 ) also hold for T t (x, y) = T [1] t (x 1 , y 1 ) · T [2] t (x 2 , y 2 ), together with Q = Q 1 Q 2 . This is enough, since by simple induction we shall get that in the general case T t (x, y) = T [1] t (x 1 , , and, consequently, the assumptions of Theorem A will be fulfilled. To prove the claim let T [ j] t (x j , y j ) and Q j satisfy (A 0 )-(A 2 ) with γ j for j = 1, 2. Let 0 < γ < min(γ 1 , γ 2 ) and fix δ ∈ [0, γ ). Suppose that Q Q ⊆ Q 1 × Q 2 , where Q 1 ∈ Q 1 , Q 2 ∈ Q 2 , and without loss of generality we may assume that d Q 1 ≥ d Q 2 . Hence, Q = K × Q 2 , where K ⊆ Q 1 , see Definition 1.5 and Fig. 3. Denote by z = (z 1 , z 2 ) the center of Q = K × Q 2 . Obviously, (A 0 ) for the product follows from (A 0 ) for the factors.

Remark 3.10
It is worth to notice, that in the proof of (A 2 ) for the subordinate semigroup K t,ν we needed (A 2 ) for T t , not only (A 2 ).

Applications
In this section for simplicity, we use the same notation T t (x, y) for the integral kernels of semigroups generated by different operators.
Since |x − y| ≥ Cd Q and δ < 1/2, we have In order to estimate I 2 we consider two cases depending on the localization of Q.