T1 Conditions in Multiparameter Singular Integrals

• T1 conditions are a set of criteria defining boundedness and regularity for multiparameter singular integral operators in product spaces.
• They are characterized by two formulations—the classical vector-valued and the mixed/biparameter approach—with specific size, Hölder, and cancellation conditions.
• Under L2-boundedness, both formulations guarantee that T1 and T*1 lie in BMO, ensuring essential stability for harmonic analysis and PDE applications.

T1 conditions constitute a fundamental set of criteria in the theory of multiparameter singular integral operators, particularly within the context of product spaces such as $\mathbb{R}^n \times \mathbb{R}^m$. These conditions provide the necessary framework for establishing boundedness and regularity properties of such operators—generalizing the classical Calderón-Zygmund theory to the multiparameter setting. Two principal formulations exist: the classical (vector-valued Calderón–Zygmund) approach, originating with Journé, and the mixed/“biparameter” (Pott–Villarroya–Martikainen) approach. The essential conclusion is that for $L^2$-bounded operators, both formulations identify the same class of operators, and both imply that $T1, T^*1 \in \mathrm{BMO}$ in the biparameter sense (Herrán, 2014).

1. Classical (Journé) and Mixed (Pott–Villarroya/Martikainen) Formulations

The classical Journé formulation describes a multiparameter singular integral operator $T$ via vector-valued Calderón–Zygmund theory: $$T : C_0^\infty(\mathbb{R}^n) \otimes C_0^\infty(\mathbb{R}^m) \to (C_0^\infty(\mathbb{R}^n) \otimes C_0^\infty(\mathbb{R}^m))'$$ with a distributional kernel $K(x, y)$, where $x, y \in \mathbb{R}^{n+m}$. The singularity structure is richer than the classical (single-parameter) setting, as singularities now occur along “product diagonals,” not just at $x = y$.

The kernel admits factorizations corresponding to parameter directions. For $f = f_1 \otimes f_2$, $g = g_1 \otimes g_2$,

$$\langle g, T(f) \rangle = \int_{\mathbb{R}^n} g_1(x_1) \left\langle g_2, K_1(x_1, y_1) f_2 \right\rangle f_1(y_1) dx_1 dy_1,$$

with analogous expressions in the second variable. The kernels $K_1(x_1, y_1)$ and $K_2(x_2, y_2)$ satisfy size and Hölder continuity conditions such as: $|K_1(x_1, y_1)| \leq C |x_1 - y_1|^{-d_1},$

$$|K_1(x_1, y_1) - K_1(x_1', y_1)| \leq C \frac{|x_1 - x_1'|^\delta}{|x_1 - y_1|^{d_1+\delta}}.$$

A priori boundedness assumptions on certain operator components are sometimes required due to the vector-valued nature of the estimates.

In contrast, the mixed/biparameter formulation (Pott–Villarroya/Martikainen) eschews vector-valued kernels and instead imposes “mixed” size, Hölder, and mixed Hölder–size estimates directly on the scalar kernel: $$|K(x, y)| \leq C \frac{1}{|x_1 - y_1|^n |x_2 - y_2|^m}$$

$$|K(x, y) - K((x_1', x_2), y)| \leq C \frac{|x_1 - x_1'|^{\delta}}{|x_1 - y_1|^{n+\delta}|x_2 - y_2|^m}$$

Additional conditions include:

• “Separated” or “mixed” Hölder continuity and cancellation conditions
• A weak boundedness property (WBP):

$$|\langle T(\chi_Q \otimes \chi_V), \chi_Q \otimes \chi_V \rangle| \leq C |Q| |V|$$

• “Diagonal” BMO (bounded mean oscillation) conditions controlling the action of $T$ and its adjoint on bump functions adapted to cubes.

2. The Role and Structure of T1 Conditions

T1 conditions impose regularity/cancellation properties of $T$ (and $T^*$) on the constant function $1$. In the Journé setting, for $L^2$-bounded bi-parameter singular integrals, these take the form: $$T1,\, T^*1 \in \mathrm{BMO} \text{ (bi-parameter sense)}$$ They are verified by testing the kernels using adapted bump functions with mean zero and ensuring uniform control over the resulting actions, e.g.: $|\langle K(x, y) \varphi, \psi \rangle| \leq C$ uniformly in test functions $\varphi, \psi$ (see Equation 2.10).

In the mixed formulation, T1 conditions are incorporated within the weak boundedness property and BMO-type testing via cube-adapted functions: $$\left| \langle T(a_Q \otimes \chi_V), \chi_Q \otimes a_V \rangle \right| \leq C |Q||V|$$ for $a_Q, a_V$ mean-zero on $Q$ and $V$, respectively.

In both approaches, these conditions are central in establishing the validity of a T1-type theorem in the multiparameter context, ensuring boundedness and BMO regularity of $T1$ and $T^*1$.

3. Equivalence of Classical and Mixed Type Conditions for $L^2$-Bounded Operators

A principal result is that for $T$ extending to a bounded operator on $L^2(\mathbb{R}^d)$, the classical (Journé) and mixed (Pott–Villarroya/Martikainen) T1 conditions are equivalent: each set of conditions is sufficient to guarantee that the other holds. That is,

• If $T$ satisfies the mixed type (Pott–Villarroya/Martikainen) conditions and is $L^2$-bounded, then there exist Calderón–Zygmund kernels satisfying all standard conditions of the classical approach (Sections 4, ‘Mixed Type Conditions Imply Classical Conditions’).
• Conversely, if $T$ satisfies the classical Journé conditions together with the classical WBP and is $L^2$-bounded, then all the required mixed type kernel, Hölder, and cancellation conditions hold (Section 5, ‘Classical Conditions Imply Mixed Type Conditions’).

Thus, the operator classes defined by both frameworks coincide under $L^2$-boundedness. The upshot is that validation of either set of conditions suffices in establishing all core properties, including $T1,\, T^*1\in \mathrm{BMO}$.

4. Canonical Kernel Estimates and Conditions

Below is a summary of fundamental kernel conditions appearing in both frameworks.

Condition	Classical (Journé)	Mixed (Pott–Villarroya/Martikainen)
Product size	$|K(x, y)| \leq C|x-y|^{-d}$ (scalar)	$|K(x, y)| \leq C/|x_1-y_1|^n |x_2-y_2|^m$
Product Hölder	$$|K(x,y) - K(x',y)| \leq C\frac{|x-x'|^\delta}{|x-y|^{d+\delta}}$$	$$|K(x,y) - K((x_1', x_2), y)| \leq C\frac{|x_1-x_1'|^{\delta}}{|x_1-y_1|^{n+\delta}|x_2-y_2|^m}$$
Mixed weak boundedness	Classical WBP (vector-valued)	$$|\langle T(\chi_Q \otimes \chi_V),\chi_Q \otimes \chi_V\rangle| \leq C|Q||V|$$
“Diagonal” BMO	Tested via bump functions (via adjoints)	$$|\langle T(a_Q \otimes \chi_V), \chi_Q \otimes a_V\rangle| \leq C|Q||V|$$

Each condition targets a particular direction of regularity (single variable or mixed), and defines either the local singularity structure or the behavior under adapted testing.

5. Practical Implications and Consequences

The equivalence of the two conditions presents several advantages:

• Verification: Mixed type conditions are sometimes more tractable in concrete applications because they involve direct kernel testing with bump or indicator functions; this simplifies analysis compared to verifying vector-valued conditions.
• Boundedness and BMO: Either formulation, when combined with $L^2$-boundedness, guarantees that $T1, T^*1 \in \mathrm{BMO}$ in the biparameter sense and that all relevant weak boundedness properties hold.
• Flexibility: The mixed conditions, not relying on a priori vector-valued boundedness, extend applicability to cases where such properties are difficult or impractical to establish.
• Corollary 4.6: Any operator $T$ defined by the mixed type kernel conditions and $L^2$-bounded automatically satisfies the classical weak boundedness property and BMO inclusion for $T1$ and $T^*1$.

A plausible implication is that, for applications involving product-space singular integrals, analysts may strategically choose the simpler-to-verify mixed conditions confident that the full classical theory applies.

6. Summary Table: Classical vs. Mixed Type T1 Conditions (under $L^2$ boundedness)

Formulation	Kernel Conditions	Testing/Weak Bounds	BMO Regularity Outcome
Journé (classical)	Vector/calderón–zygmund on $(K_1, K_2)$	WBP, cancellation (vector-valued)	$T1, T^*1\in\mathrm{BMO}$ (biparameter)
Mixed (Pott–Villarroya/Martikainen)	Size, mixed and separated Hölder, mixed WBP	BMO on adapted (cube) functions	$T1, T^*1\in\mathrm{BMO}$ (biparameter)

Both define the same operator class for $L^2$-bounded $T$. Mixed conditions are often easier to use in practice, especially when “bump function” testing is more accessible than vector-valued kernel analysis.

7. Conclusions

The T1 conditions for multiparameter operators, as addressed in (Herrán, 2014), form a comprehensive and unifying principle for the analysis of bi- and multi-parameter singular integrals on product spaces. The main finding is the equivalence—under $L^2$-boundedness—of the vector-valued and mixed type formulations, which secures all standard properties (boundedness, weak boundedness, BMO regularity) essential for further harmonic analysis and PDE applications in multiparameter settings. The equivalence both simplifies and broadens practical access to the full power of the product-space T1 theorem.